Total number of papers on ( Thursday, August 28, 2025 ) is 165.
Fateme Movahedi, Ivan Gutman, Izudin Redžepović, Boris Furtula
Diminished Sombor index
MATCH Communications in Mathematical and in Computer Chemistry 95 (2026) 141-162
Abstract
The diminished Sombor index of a graph \( G \) is defined as
\[
DSO(G)=\sum \frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},
\]
where \( d_u \) and \( d_v \) are the degrees of vertices \( u \) and \( v \), and the summation goes over all pairs of adjacent vertices. Although \( DSO \) was introduced as early as in 2021, its properties were not studied so far. The present paper is aimed at filling this gap. We obtain bounds on \( DSO \), characterize the extremal graphs, and establish Nordhaus-Gaddum-type relations. In addition, we report results of numerical studies of the structure-dependency of \( DSO \) and its chemical applicability.
Akbar Ali, Ivan Gutman, Boris Furtula, Abeer M. Albalahi, Amjad E. Hamza
On chemical and mathematical characteristics of generalized degree-based molecular descriptors
AIMS Mathematics 10 (2025) 6788-6804
Abstract
This paper deals with the properties of the generalized Gutman–Milovanović index, generalized elliptic-Sombor index, generalized Zagreb-Sombor index, and general Euler-Sombor index. These include, as special cases, several previously studied molecular descriptors and most of their general versions; for instance, the general Randić index, the general sum-connectivity index, the general Sombor index, etc. The aforementioned descriptors are examined for their applicability in predicting 13 properties of octane isomers, and the results are compared with the ones generated by a benchmark data set (proposed by the International Academy of Mathematical Chemistry), containing 102 descriptors of octane isomers, and also with variable and discrete Adriatic indices. Although these descriptors slightly outperform the descriptors considered for comparison in several cases, a considerable improvement is detected in the case of boiling point. Several fundamental bounds and optimal results of the above-said descriptors are also reported.
Boris Furtula, Mert Sinan Oz
Geometric–quadratic index from a mathematical perspective
Iranian Journal of Mathematical Chemistry 16 (2025) 85-92
Abstract
The geometric-quadratic index (GQ) was defined in 2021 by V. R. Kulli. In a recent study, QSPR analysis for the octane isomers of GQ and some other newly defined topological indices was presented. This analysis has revealed that GQ dominates over many of the well-known topological indices in terms of chemical applicability potential, especially for the heat of vaporization. These results inspired us to investigate the mathematical properties of GQ. In this paper, extremal graphs for GQ are investigated among connected graphs, trees, and unicyclic graphs. In addition, several mathematical relations between GQ and some well-known topological indices are presented.
Ivan Gutman, Boris Furtula
Calculating vertex energies of graphs - a tutorial
MATCH Communications in Mathematical and in Computer Chemistry 93 (2025) 691-698
Abstract
In 2018, Arizmendi introduced the concept of energy of a vertex of a graph, \( \mathcal E_G(v) \), i.e., the distribution of graph energy \( \mathcal E(G) \) over the vertices of the underlying graph \( G \). We now show how \( \mathcal E_G(v) \) should actually be calculated.
Boris Furtula, Mert Sinan Oz
On properties of the frist inverse Nirmala index
Journal of Mathematical Chemistry 63 (2025) 96-104
Abstract
The first inverse Nirmala index is a novel degree-based topological descriptor that was introduced in 2021. Preliminary QSPR investigations suggest that this index deserves further consideration because of its unusually good predictive potential. This paper investigates the relations between this index with some elementary graph quantities and some related degree-based topological index. Further, the computational analysis will reveal extremal graphs among trees, molecular trees, all connected graphs, and their molecular counterparts.
Boris Furtula, Mert Sinan Oz
Complementary Topological Indices
MATCH Communications in Mathematical and in Computer Chemistry 93 (2025) 247-263
Abstract
An edge of a graph can be geometrically represented by points \( (d_r, d_s) \) and \( (d_s, d_r) \) in a 2D coordinate system, where coordinates are, obviously, the degrees of the edge's end-vertices. Recently, using such a geometrical point of view of a graph edge, a couple of topological invariants were put forward. They have attracted considerable attention among chemical graph theorists. This paper introduces a novel approach for devising "geometrical" topological indices. Finally, special attention is focused on the complementary second Zagreb index as a representative of the introduced approach.
Akbar Ali, Ivan Gutman, Boris Furtula, Izudin Redžepović, Tomislav Došlić, Zahid Raza
Extremal results and bounds for atom-bond sum-connectivity index
MATCH Communications in Mathematical and in Computer Chemistry 92 (2024) 271-314
Abstract
The \( ABS \) (atom-bond sum-connectivity) index is a topological index, that was introduced in 2022 by amalgamating the main ideas of two well-examined indices. Mathematical aspects (especially, extremal results and bounds) of the \( ABS \) index have already been studied considerably. The primary goal of this review paper is to collect known bounds and extremal results regarding the \( ABS \) index. Several new extremal results, which follow easily from existing general results, are also given. Moreover, a number of open problems and conjectures, arising from the reported results, are proposed.
Ivan Gutman, Boris Furtula, Mert Sinan Oz
Geometric approach to vertex-degree-based topological indices-Elliptic Sombor index, theory and application
International Journal of Quantum Chemistry 124 (2024) e27346-(10 pp.)
Abstract
A novel geometric method is proposed for constructing vertex-degree-based molecular structure descriptors (topological indices). The model is based on an ellipse whose focal points represent the degrees of a pair of adjacent vertices. This approach enables a geometric interpretation of several previously known topological indices, and lead to design of a few new. The area of the ellipse induces a vertex-degree-based topological index of remarkable simplicity, which we call elliptic Sombor index(\( ESO \)). The main mathematical properties of \( ESO \) are established, especially its relations to other, earlier known, indices. Then the applicative potential of \( ESO \) is analyzed.
Ivan Gutman, Boris Furtula, Izudin Redžepović
On Topological Indices and Their Reciprocals
MATCH Communications in Mathematical and in Computer Chemistry 91 (2024) 287-297
Abstract
If \( TI(G) = \sum_\gamma F(\gamma) \) is a topological index of the graph \( G \),
then \( RTI(G) = \sum_\gamma \frac{1}{F(\gamma)} \) is the respective reciprocal index. In
contemporary mathematical chemistry, a large number of pairs (\( TI, RTI \)) have been
separately introduced and studied, but their mutual relations eluded attention. In this paper,
we determine some basic relations between \( TI \) and \( RTI \), and then focus our attention
to the pair Wiener index - Harary index.
If \( G \) is a connected graph and \( d(u, v) \) the distance between its vertices \( u \)
and \( v \), then the Wiener and Harary indices are \( W = \sum_{u,v} d(u,v) \) and \( W = \sum_{u,v} \frac{1}{d(u,v)} \) , respectively. In this paper the product \( W \cdot H \) is studied. The minimum value of \( W \cdot H \) is determined for general connected graphs and conjectured for trees. The maximum value is discussed, based on our computer-aided findings.
Ivan Gutman, Izudin Redžepović, Boris Furtula
On the product of Sombor and modified Sombor indices
Open Journal of Discrete Applied Mathematics 6 (2023) 1-6
Abstract
The Sombor index (\( SO \)) and the modified Sombor index (\( ^mSO \)) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product \( SO \cdot ^mSO \) and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either \( SO \) or \( ^mSO \). Lower and upper bounds for \( SO \cdot ^mSO \) are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product \( SO \cdot ^mSO \) is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point.
V. R. Kulli, Ivan Gutman, Boris Furtula, Izudin Redžepović
Sombor, Nirmala, Dharwad, and F-Sombor Indices - A Comparative Study
SSRG International Journal of Applied Chemistry 10 (2023) 7-10
Abstract
Let \( G \) be a molecular graph, \( d(u) \) the degree (number of first neighbors) of its vertex \( u \), and \( uv \) its edge connecting the vertices \( u \) and \( v \). In the recent literature, four vertex-degree-based molecular structure descriptors have been put forward, named Nirmala, Sombor, Dharwad, and F-Sombor indices, equal to the sum of the terms \( \sqrt{d(u)^2 + d(v)^2} \) over all edges \( uv \) of \( G \) for \( k=1,2,3, \) and 4. In this work, we compare these four indices. We find that they are highly correlated, and their value for QSPR applications is nearly the same. Therefore, only one (any one) of these topological indices should be used. From an applicative point of view, there was no need to introduce and study them separately.
Izudin Redžepović, Slađana Đorđević, Simon Brezovnik, Niko Tratnik, Petra Žigert Pleteršek, Boris Furtula, Slavko Radenković
Partition of topological indices of benzenoid hydrocarbons into ring contributions
International Journal of Quantum Chemistry 123 (2023) e27108-(10 pp.)
Abstract
This work presents a simple method for partitioning the bond-additive and atoms-pair-additive distance-based topological indices of plane graphs into the sum of contributions of inner faces. The proposed method is applied to decompose several topological indices (Wiener, hyper-Wiener, Tratch-Stankevich-Zefirov, Balaban, and Szeged indices) into the ring contributions for a series of benzenoid systems. It was found that the employed ring partitioning scheme is providing an accurate assessment of the dominant cyclic conjugation modes in the studied benzenoid hydrocarbons. Thus, the proposed method can be used as the alternative for the quantum-chemistry-based aromaticity indices which are significantly more computationally demanding.
Akbar Ali, Boris Furtula, Izudin Redžepović, Ivan Gutman
Atom‑bond sum‑connectivity index
Journal of Mathematical Chemistry 60 (2022) 2081-2093
Abstract
The branching index (also known as the connectivity index), introduced in Milan Randić’s seminal paper (J Am Chem Soc 97(23):6609–6615, 1975), is one of the the most famous, investigated, and applied among the graph-theoretical molecular descriptors. The atom-bond connectivity (\( ABC \)) index (Estrada et al in Indian J Chem A 37:849-855, 1998) and the sum-connectivity (\( SC \)) index (Zhou and Trinajstić in J Math Chem 46:1252-1270, 2009) belong to the class of successful variants of the connectivity index. In the present paper, by amalgamating the core idea of the SC and ABC indices, a new molecular descriptor is put forward-the atom-bond sum-connectivity (\( ABS \)) index. The graphs attaining the extreme values of the \( ABS \) index are determined over the classes of (molecular) trees and general graphs of a fixed order. A noteworthy property of the \( ABC \) index is that it increases when a non-isolated edge is inserted between any two non-adjacent vertices. It is proved that this property holds also for the \( ABS \) index.
Izudin Redžepović, Boris Furtula
Chemical similarity of molecules with physiological response
Molecular Diversity 27 (2023) 1603-1612
Abstract
Measuring the similarity among molecules is an important task in various chemically oriented problems. This elusive concept is hard to define and quantify. Moreover, the complexity of the problem is elevated by bifurcating the notion of molecular similarity to structural and chemical similarity. While the structural similarity of molecules is being extensively researched, the so-called chemical similarity is being mentioned scarcely. Here, we propose a way of converting the physico-chemical properties into molecular fingerprints. Then, using the apparatus of measuring the structural similarity, the chemical similarity can be assessed. The proof of a concept is demonstrated on a set of molecules that induce diverse physiological responses.
Bojana Borovićanin, Boris Furtula, Marija Jerotijević
On the minimum Harary index of graphs with a given diameter or independence number
Discrete Applied Mathematics 320 (2022) 331-345
Abstract
The graphs with diameter equal to \( n − c \), for \( 1 \leqslant c \leqslant 4 \), which attain the minimum value with respect to the Harary index are being considered. Additionally, the sharp lower bound on Harary index of connected graphs with independence number equal to \( n − c\), for \( 1 \leqslant c \leqslant 4 \) is derived. The extremal graphs are also characterized.
Boris Furtula
Trinajstić index
Discrete Mathematics Letters 9 (2022) 100-106
Abstract
Professor Trinajstić devoted years of research to deepen the knowledge of the distance–based topological indices. He was especially interested in so-called Szeged-type indices. Several such indices were introduced directly by himself, but none of them was named after him. In this paper, a novel topological invariant of this kind is proposed, and it is boldly named the Trinajstić index. The performed computational tests are justifying the introduction of this novel topological index.
Slavko Radenković, Izudin Redžepović, Slađana Đorđević, Boris Furtula, Niko Tratnik, Petra Žigert Pleteršek
Relating vibrational energy with Kekulé- and Clar-structure-based parameters
International Journal of Quantum Chemistry 122 (2022) e26867-(7 pp.)
Abstract
For all possible catacondensed Kekuléan molecules having four, five, and six hexagons,the molecular vibrational energies were calculated within the harmonic approximationat the HF, B3LYP, and M06-2X levels of theory in combination with the 6-311G(d,p)basis set. The obtained vibrational energies were found to be a linear function of the Kekulé structure count \(K\) within the sets of isomeric molecules. By employing the recently introduced generalized Zhang-Zhang polynomial, it was shown that the molecular vibrational energies can be related to the Clar-structure-based parameters. The obtained approximate formulas can accurately reproduce the vibrational energies with an average absolute error less than 1 kJ/mol. In addition, these formulas can providefurther details on the structural dependence of molecular vibrational energies.
Zhao Wang, Yaping Mao, Yue Li, Boris Furtula
On relations between Sombor and other degree-based indices
Journal of Applied Mathematics and Computing 68 (2022) 1-17
Abstract
Recently, a novel topological invariant based on degrees of end-vertices of an edge in a graph was put forward. It was named Sombor index. The definition of the Sombor invariant suggests another, rather a geometric view onto graph edges. Here, the mathematical relations between the Sombor index and some other well-known degree-based descriptors are investigated. Then, a few results of Nordhaus-Gaddum-type are obtained. Finally, computational testing and comparison with other well-established indices are presented.
Boris Furtula, Slavko Radenković, Izudin Redžepović, Niko Tratnik, Petra Žigert Pleteršek
The generalized Zhang-Zhang polynomial of benzenoid systems - theory and applications
Applied Mathematics and Computation 418 (2022) 126822-(14 pp.)
Abstract
The generalized Zhang-Zhang (\(GZZ\)) polynomial was introduced recently aiming to increase the sensitivity of the well-known Zhang–Zhang polynomial onto \(\pi\)-electron cyclic conjugation of 10-membered rings. Here, the recursive formulas for the calculation of the \(GZZ\) of the benzenoid systems are derived. Then, an algorithm for calculating the \(GZZ\) of benzenoid chains is given. Lastly, testing the chemical applicability of \(GZZ\) is performed.
Ivan Gutman, Izudin Redžepović, Boris Furtula, Ali Mohammed Sahal
Energy of graphs with self-loops
MATCH Communications in Mathematical and in Computer Chemistry 87 (2022) 645-652
Abstract
The energy of graphs containing self-loops is considered. If the graph \(G\) of order \(n\) contains \(\sigma\) self-loops, then its energy is defined as \(E(G) = \sum |\lambda_i − \sigma/n|\) where \(\lambda_1, \lambda_2,\dots, \lambda_n\) are the eigenvalues of the adjacency matrix of \(G\). Some basic properties of \(E(G)\) are established, and several open problems pointed out or conjectured.
Svetlana Marković, Izudin Redžepović, Boris Furtula
Dependence of the enthalpy of formation of phenols on molecular structure - semiempirical study
Polycyclic Aromatic Compounds 41 (2021) 1755-1766
Abstract
The ability of three semiempirical methods (PM3, PM5, and PM7) in reproducing the enthalpy of formation (\(\Delta_f H^\circ\)) of polycyclic aromatic phenols (PAPs) was tested by comparing experimental and calculated values for 33 compounds. In addition, the results from the semiempirical methods were compared to those obtained from the suitable isodesmic reactions, performed using the B3LYP–D3 and M06–2X functionals in combination with the 6-311 + G(d,p) basis set. With an average absolute error less than \(10 kJ mol^{−1}\), all five methodologies yielded mutually comparable results. Considering low computational cost of semiempirical methods, the influence of structural features on \(\Delta_f H^\circ\) of PAPs was investigated using the PM5 and PM7 methods. It was found that the greatest influence on \(\Delta_f H^\circ\) has the size of the molecules, while the effects of other properties, i.e., the position of the OH group, number of bay regions, and molecular branching are weaker. The influence of the OH group position decreases with its moving away from molecular ends, and eventually becomes insignificant. As branching and the number of bay regions are mutually dependent and respective molecules are often nonplanar, it is difficult to examine individual contributions of these two structural features. In general, \(\Delta_f H^\circ\) decreases with increasing number of bay regions and molecular branching.
Yaping Mao, Boris Furtula
Steiner distance in chemical graph theory
MATCH Communications in Mathematical and in Computer Chemistry 86 (2021) 211-287
Abstract
Steiner distance \(d_G(S)\) is a natural generalization of the concept of distance in a graph. For a connected graph \(G\) of order at least 2 and \(S \subseteq V(G)\), \(d_G(S)\) is equal to the minimum size among all connected subgraphs whose vertex sets are equal to the set \(S\). Here, the known results on the Steiner distance parameters used in chemical graph theory such as Steiner Wiener index, Steiner degree distance, Steiner Harary index, Steiner Gutman index, Steiner hyper-Wiener index, and Steiner Hosoya polynomial are surveyed. Additionally, some conjectures and open problems are listed.
Izudin Redžepović, Slavko Radenković, Boris Furtula
Effect of a ring onto values of eigenvalue-based molecular descriptors
Symmetry 13 (2021) 1515-(11 pp.)
Abstract
The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they are being ordinarily employed for predicting physico–chemical properties and/or biological activities of molecules. It has been shown that these indices describe well molecular features that are depending on fine structural details. Therefore, revealing the impact of structural details on the values of the eigenvalue–based topological indices should give a hunch how physico–chemical properties depend on them as well. Here, an effect of a ring in a molecule on the values of the graph energy, Estrada index and the resolvent energy of a graph is examined.
Zhao Wang, Yaping Mao, Boris Furtula, Xu Wang
Bounds for the spectral radius and energy of extended adjacency matrix of graphs
Linear and Multilinear Algebra 69 (2021) 1813-1824
Abstract
An extend adjacency matrix of a graph (\(A_{ex}\)) was introduced decades ago as a precursor for developing a few quite useful molecular topological descriptors. The spectral radius (\(\eta_1\)) of the extended adjacency matrix and the extended energy of a graph (\(\mathcal{E}_{ex}\)) have been successfully utilized in QSPR/QSAR investigations. Here, the \(\eta_1\) and \(\mathcal{E}_{ex}\) have been further mathematically analyzed. Several sharp upper bounds for the \(\mathcal{E}_{ex}\) are obtained. In addition, the Nordhaus–Gaddum-type results for the \(\eta_1\) and \(\mathcal{E}_{ex}\) are presented.
Akbar Ali, Kinkar C. Das, Darko Dimitrov, Boris Furtula
Atom–bond connectivity index of graphs: A review over extremal results and bounds
Discrete Mathematics Letters 5 (2021) 68-93
Abstract
The atom–bond connectivity (\(ABC\)) index was introduced in the last quarter of the 1990s to improve the prediction power of the Randić index. Later on, in 2008, the factor \(\sqrt{2}\) was dropped from the original definition of the \(ABC\) index, and some additional chemical applications of this index were reported, which resulted in considerable interest in studying the mathematical properties of the \(ABC\) index. There are more than a hundred papers devoted to the mathematical aspects of this graph invariant. The primary purpose of this review is to gather the existing bounds and extremal results concerning the \(ABC\) index.
Muhuo Liu, Kun Cheng, Boris Furtula
Minimum augmented Zagreb index of \(c\)-cyclic graphs
Discrete Applied Mathematics 295 (2021) 32-38
Abstract
The augmented Zagreb index (\(AZI\)) has proved its applicability potential in various chemical researches. Its mathematical features are being investigated vigorously. Here, the unique graph that minimizes the \(AZI\) among all \(c\)-cyclic graphs with fixed order is characterized.
Izudin Redžepović, Boris Furtula
Comparative study on structural sensitivity of eigenvalue-based molecular descriptors
Journal of Mathematical Chemistry 59 (2021) 476-487
Abstract
Structural sensitivity is one of the most important and the least investigated property of the topological molecular descriptors. This paper reports results on the structural sensitivity of graph energy, Estrada index, and resolvent energy on several series of catacondensed and pericondensed isomeric benzenoid hydrocarbons. Recently, a novel method for assessing the structural sensitivity of topological molecular descriptors was put forward, which is applied here. This algorithm consists of several different steps and it is based on Tanimoto index and Morgan circular fingerprints. It was found that graph energy, Estrada index, and resolvent energy exert similar structural sensitivity on catacondensed isomers. The graph energy showed the best performance on pericondensed molecules. Additionally, the sensitivities of these descriptors were tested on the catacondensed isomers with the increasing number of bays, coves, and fjords. It was revealed that these descriptors gradually change with the increasing number of these structural features. The Estrada index and resolvent energy perform similarly and in some cases with the same structural sensitivity. This may be attributed to the high correlation between them. The graph energy showed superiority over the Estrada index and resolvent energy on pericondensed isomers.
Akbar Ali, Boris Furtula, Ivan Gutman, Damir Vukičević
Augmented Zagreb index: Extremal results and bounds
MATCH Communications in Mathematical and in Computer Chemistry 85 (2021) 211-244
Abstract
The augmented Zagreb index (\(AZI\)) is a molecular structure descriptor introduced about a decade ago. Chemical applicability of \(AZI\) was tested in several studies, where it was found that in most cases \(AZI\) outperforms other structure descriptors of this type. This survey paper outlines extremal results and bounds on \(AZI\) that have been reported until now.
Saša Vujošević, Goran Popivoda, Žana Kovijanić Vukićević, Boris Furtula, Riste Škrekovski
Arithmetic-geometric index and its relations with geometric-arithmetic index
Applied Mathematics and Computation 391 (2021) 125706-(13 pp.)
Abstract
The arithmetic–geometric index (\(AG(G)\)) was recently introduced as a modification of the well-known geometric–arithmetic index (\(GA(G)\)). This paper reports results on searching for extremal \(AG\)-graphs for various classes of simple graphs. Additionally, relations between these two indices are elaborated. Results on combinations \(AG + GA\), \(AG - GA\), \(AG \cdot GA\), and \(AG/GA\) are given. The paper is concluded with four conjectures that have been derived based on computer investigations.
Biljana Arsić, Boris Furtula, Marjan Ranđelović
Evaluation of the constructed 3D models of RNAs: A review
Facta Universitatis, Series Physics, Chemistry and Technology 18 (2020) 39-45
Abstract
The development of new experimental techniques, such as cryo-electron spectroscopy, enables insight into the structural features inside cells. However, in specific cases, it is still not possible to get the cryo images. Therefore, the development of the scores for the evaluation of the quality of the constructed RNAs, similarly to the proteins, is a prerequisite for the investigation of the diseases caused by the organisms not well investigated. Here, we are providing a summary of the evaluation scores in use for the prediction of the quality of the constructed 3D models of the RNAs.
Izudin Redžepović, Boris Furtula
Predictive potential of eigenvalue-based topological molecular descriptors
Journal of Computer-Aided Molecular Design 34 (2020) 975-982
Abstract
This study is directed toward assessing the predictive potential of eigenvalue-based topological molecular descriptors. The graph energy, Estrada index, resolvent energy, and the Laplacian energy were tested as parameters for the prediction of boiling points, heats of formation, and octanol/water partition coefficients of alkanes. It was shown that an eigenvalue-based molecular descriptor cannot be individually used for successful prediction of these physico-chemical properties, but the first Zagreb index, the number of zeros in the spectrum and the number of methyl groups must be also involved in the models. Performed statistics show that the models constructed using the Estrada index and resolvent energy are significantly better than ones with the energy of a graph and the Laplacian energy. Such a trend is even more noticeable in the case of octanol/water partition coefficients of alkanes.
Izudin Redžepović, Boris Furtula
On degeneracy of A-eigenvalue–based molecular descriptors and r-equienergetic chemical trees
MATCH Communications in Mathematical and in Computer Chemistry 84 (2020) 385-397
Abstract
Among more than two hundreds of eigenvalue-based topological indices only a couple of them are defined using the eigenvalues devised from the adjacency matrix of a graph. The resolvent energy is probably the most recent-one such an index. In this article, the degeneracy of the energy, Estrada index, and the resolvent energy is presented. The specious degeneracy of the resolvent energy in the case of chemical trees is discussed. Then, the data on searching for resolvent equienergetic chemical trees is given.
Izudin Redžepović, Boris Furtula
Resolvent energy and Estrada index of benzenoid hydrocarbons
Journal of the Serbian Society for Computational Mechanics special issue (2020) 37-44
Abstract
The relationship between the resolvent and Estrada indices for the alkanes has been recently demonstrated. That relationship involved the first Zagreb index besides these two eigenvalue-based molecular descriptors. In this paper, the quality of the correlation is tested in the case of isomeric benzenoid hydrocarbons, where the first Zagreb index is constant. Extraordinary linear correlations are identified for all studied groups of isomeric benzenoid hydrocarbons. Additionally, the relationship of these indices with the boiling points of a set of benzenoid hydrocarbons is presented.
Izudin Redžepović, Boris Furtula, Ivan Gutman
Relating total \(\pi\)-electron energy of benzenoid hydrocarbons with HOMO and LOMO energies
MATCH Communications in Mathematical and in Computer Chemistry 84 (2020) 229-237
Abstract
Within the Hückel molecular orbital model, the total \(\pi\)-electron energy, \(E_\pi\), the highest occupied molecular orbital (\(HOMO\)) energy, \(E_{HOMO}\), and the lowest occupied molecular orbital (\(LOMO\)) energy, \(E_{LOMO}\), can be expressed in terms of eigenvalues of the adjacency matrix of the underlying molecular graph. In this paper, relations between \(E_\pi\), \(E_{HOMO}\), and \(E_{LOMO}\) are examined. Approximate expressions are established, relating \(E_\pi\) with \(E_{HOMO}\) and \(E_{LOMO}\) in the case of benzenoid hydrocarbons.
Risto Atanasov, Boris Furtula, Riste Škrekovski
Trees with minimum weighted Szeged index are of a large diameter
Symmetry 12 (2020) 793-(10 pp.)
Abstract
The weighted Szeged index (\(wSz\)) has gained considerable attention recently because of its unusual mathematical properties. Searching for a tree (or trees) that minimizes the \(wSz\) is still going on. Several structural details of a minimal tree were described. Here, it is shown a surprising property that these trees have maximum degree at most 16, and as a consequence, we promptly conclude that these trees are of large diameter.
Izudin Redžepović, Yaping Mao, Zhao Wang, Boris Furtula
Steiner degree distance indices: Chemical applicability and bounds
International Journal of Quantum Chemistry 120 (2020) e26209-(9 pp.)
Abstract
The k-center Steiner degree distance (\(SDD_k(G)\)) has recently been introduced as a natural extension of the degree distance \(DD(G)\). In this paper, the prediction potential of \(SDD_k(G)\) is discussed. Then, the relation between this and some other well-known distance-based indices of trees is derived to explain its prediction potential. Finally, the lower and upper bounds of \(SDD_k(G)\) in terms of some other graph invariants are presented.
Izudin Redžepović, Boris Furtula
On relationships of eigenvalue–based topological molecular descriptors
Acta Chimica Slovenica 67 (2020) 312-318
Abstract
Three eigenvalue-based topological molecular descriptors are compared using several datasets of alkanes. Two of them are well-known and frequently employed in various QSPR/QSAR investigations, and third-one is a newly derived whose predictive potential is yet to be proven. The relations among them are found and discussed. Structural parameters that govern these relations are identified and the corresponding formulas based on multiple linear regression have been obtained. It has been shown that all three investigated indices are encoding almost the same structural information of a molecule. They differ only by the extent of the sensitivity on a structural branching of a molecule and on the number of non-bonding molecular orbitals.
Marija Antić, Slađana Đorđević, Boris Furtula, Slavko Radenković
Magnetically induced current density in non-planar fully benzenoid hydrocarbons
The Journal of Physical Chemistry A 124 (2020) 371-378
Abstract
In our recent paper, the effects of molecular planarity on the local aromaticity in several series of increasingly planar fully benzenoid hydrocarbons were examined. It was found that the Clar formulas can provide correct information on the local aromaticity distribution even in nonplanar fully benzenoid systems. In the present work, the influence of molecular planarity on the ab initio magnetically induced current densities was examined for the same sets of molecules. The planarity effects were rationalized by examining the origins of the induced current density through the virtual transitions between occupied and unoccupied molecular orbitals.
Ivan Gutman, Izudin Redžepović, Boris Furtula
Two stability criteria for benzenoid hydrocarbons and their relation
Croatica Chemica Acta 92 (2019) 473-475
Abstract
A new, simple, relation is established between the total \(\pi\)-electron energy and the HOMO-LUMO gap, applicable to benzenoid hydrocarbons.
Izudin Redžepović, Svetlana Marković, Boris Furtula
On structural dependence of enthalpy of formation of catacondensed benzenoid hydrocarbons
MATCH Communications in Mathematical and in Computer Chemistry 82 (2019) 663-678
Abstract
Dependence of the enthalpy of formation (\(\Delta H_f\)) of catacondensed benzenoid hydrocarbons (CBHs) on structural features was examined. To elucidate the influence of the molecular size (expressed through the number of hexagons, \(h\)), number of bays (\(B\)), number of coves (\(C\)), number of fjords (\(F\)), and molecular branching (expressed through the number of the \(A_3\)-type hexagons, \(h_{A_3}\)) on \(\Delta H_f\), a simple mathematical model was developed. Namely, \(\Delta H_f\) of CBHs was approximated as a linear combination of first several spectral moments, up to \(M_{12}\). For this purpose, multiple linear regression was applied, where the \(\Delta H_f\) values obtained from the PM7 calculations for 1221 randomly chosen CBHs were used as learning set. Fortunately, the formulas for these spectral moments that depend on molecular structure have already been derived, implying that the model describes \(\Delta H_f\) in terms of structural details of CBHs. Agreement between the experimental and calculated \(\Delta H_f\) is satisfactory, with an average relative error of 4.5%. It was found that the major part of \(\Delta H_f\) is determined by h, where \(\Delta H_f\) increases with increasing h. Subtle variations in the value of \(\Delta H_f\) are explained by other structural features of a molecule. \(\Delta H_f\) decreases with increasing B, C, and F, but increases with increasing \(h_{A_3}\). Contribution of each structural property was quantitatively determined. This is the first study that describes \(\Delta H_f\) of CBHs in terms of structural features that can be straightforwardly obtained.
Boris Furtula, Ivan Gutman, Marjan Matejić, Emina Milovanović, Igor Milovanović
Some new lower bounds for augmented Zagreb index
Journal of Applied Mathematics and Computing 61 (2019) 404-415
Abstract
Let \(G=(V,E), V=\{1,2,\ldots,n\}\), be a simple connected graph with \(n\geq 3\) vertices and \(m\) edges, with vertex degree sequence \(d_1 \geq d_2 \geq \cdots \geq d_n, d_i = d(i)\). The augmented Zagreb index is defined as \(AZI = \sum_{i\sim j} \left(\frac{d_i\,d_j}{d_i + d_j - 2}\right)^3\), where \(i\sim j\) denotes adjacency of vertices \(i\) and \(j\). Some new lower bounds for \(AZI\) are obtained.
Marija Rakić, Boris Furtula
A novel method for measuring the structure sensitivity of molecular descriptors
Journal of Chemometrics 33 (2019) e3138-(10 pp.)
Abstract
A response of a molecular descriptor on subtle structural changes is a parameter that shed light on its quality. Until now, there was a just one attempt to quantify this property. Here, a novel method for calculating this parameter, based on fingerprint molecular similarity, is given. Preliminary investigations show that this new method outperforms already existing one. In addition, some well-established degree-, distance-, and eigenvalue-based topological molecular descriptors are subjected to this test, using acyclic, unicyclic, bicyclic, and tricyclic decanes and sorted accordingly.
Jan Bok, Boris Furtula, Nikola Jedličková, Riste Škrekovski
On extremal graphs of weighted Szeged index
MATCH Communications in Mathematical and in Computer Chemistry 82 (2019) 93-109
Abstract
An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index (\(wSz(G)\)). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal \(wSz(G)\). Finding a tree with the minimal \(wSz(G)\) is not an easy task to be done. Here, we present the minimal trees up to 25 vertices obtained by computer and describe the regularities which retain in them. Our preliminary computer tests suggest that a tree with the minimal \(wSz(G)\) is also the connected graph of the given order that attains the minimal weighted Szeged index. Additionally, it is proven that among the bipartite connected graphs the complete balanced bipartite graph \(K_{\left\lfloor n/2\right\rfloor\left\lceil n/2 \right\rceil}\) attains the maximal \(wSz(G)\). We believe that the \(K_{\left\lfloor n/2\right\rfloor\left\lceil n/2 \right\rceil}\) is a connected graph of given order that attains the maximum \(wSz(G)\).
Ana Gligorijević, Svetlana Marković, Izudin Redžepović, Boris Furtula
Application of spectral graph theory on the enthalpy change of formation of acyclic saturated ketones
Journal of the Serbian Chemical Society 83 (2018) 1339-1349
Abstract
The dependence of the enthalpy change of formation of saturated acyclic ketones on molecular structure (the number of carbon atoms, the position of the carbonyl group, and the branching of the molecules) was investigated. For this purpose, a simple computational model, the parameterization of which is based on spectral graph theory, was developed. It was found that the major part of the enthalpy change of formation is determined by molecular size, whereas the fine structure of the enthalpy change of formation is determined by the branching of the molecule and the position of the carbonyl group. The developed model proved itself very useful for such investigations. The model is simple and practical, and the agreement between the experimental and calculated enthalpy changes of formation is very good, with an average relative error of 0.7%.
Ivan Gutman, Boris Furtula, Vladimir Katanić
Randić index and information
AKCE International Journal of Graphs and Combinatorics 15 (2018) 307-312
Abstract
The Randić index \(R(G)\) is one of the classical graph-based molecular structure descriptors that found countless applications in chemistry and pharmacology. The mathematical background of this index is also well elaborated. We now point out a hitherto unnoticed feature of \(R(G)\), namely its connection with the degree-based information content of a (molecular) graph. This connection is based on the linear correlation between \(R(G)\) and the logarithm of the multiplicative version of the Randić index.
Boris Furtula, Kinkar C. Das, Ivan Gutman
Comparative analysis of symmetric division deg index as potentially useful molecular descriptor
International Journal of Quantum Chemistry 118 (2018) e25659-(14 pp.)
Abstract
There are several dozens of vertex–degree–based (\(VDB\)) molecular structure descriptors currently studied and proposed to be used in quantitative structure-property/activity relationships (QSPR/QSAR) researches. Among them, just a couple are recognized as promising and worth of practical applications. One of the newest is the symmetric division deg index (\(SDD\)). This article is devoted to a thorough multifaceted analysis of \(SDD\) and its comparison with other \(VDB\) topological indices. We show that the applicative potential of \(SDD\) is comparable to already well-established \(VDB\) structure descriptors. Additionally, some of conducted tests indicate its supremacy over other \(VDB\) molecular indices.
Kinkar C. Das, Ivan Gutman, Igor Milovanović, Emina Milovanović, Boris Furtula
Degree-based energies of graphs
Linear Algebra and Its Applications 554 (2018) 185-204
Abstract
Let \(G=(V,E)\) be a simple graph of order \(n\) and size \(m\), with vertex set \(V(G) = \{v_1,v_2,\ldots,v_n\}\), without isolated vertices and sequence of vertex degrees \(\Delta = d_1 \geq d_2 \geq\cdots\geq d_n = \delta > 0\), \(d_i = d_G(v_i)\). If the vertices \(v_i\) and \(v_j\) are adjacent, we denote it as \(v_i\,v_j \in E(G)\) or \(i \sim j\). With \(TI\) we denote a topological index that can be represented as \(TI = TI(G) = \sum_{i \sim j} \mathcal{F}(d_i,d_j)\), where \(\mathcal{F}\) is an appropriately chosen function with the property \(\mathcal{F}(x,y) = \mathcal{F}(y,x)\). A general extended adjacency matrix \(A=(a_{ij})\) of \(G\) is defined as \(a_{ij} = \mathcal{F}(d_i,d_j)\) if the vertices vi and vj are adjacent, and \(a_{ij} = 0\) otherwise. Denote by \(f_i, i=1,2,\ldots,n\) the eigenvalues of \(A\). The "energy" of the general extended adjacency matrix is defined as \(E_{TI} = E_{TI}(G) = \sum_{i=1}^n |f_i|\). Lower and upper bounds on \(E_{TI}\) are obtained. By means of the present approach a plethora of earlier established results can be obtained as special cases.
Yuede Ma, Shujuan Cao, Yongtang Shi, Ivan Gutman, Matthias Dehmer, Boris Furtula
From the connectivity index to various Randić-type descriptors
MATCH Communications in Mathematical and in Computer Chemistry 80 (2018) 85-106
Abstract
The Randić index was proposed by Randić in 1975 and has been widely studied in different areas. We briefly review the Randić index and various Randić-type descriptors from 1975 to date, including higher-order Randić indices, zeroth order Randić indices, \(D-L-S\) generalization, sum-connectivity indices, geometric-arithmetic indices, harmonic index, atom-bond connectivity index, Balaban index, Randić matrix, Randić spectrum, Randić energy, etc. We also point out some important applications of the various Randić-type indices.
Ivan Gutman, Boris Furtula
The total \(\pi\)-electron energy saga
Croatica Chemica Acta 90 (2017) 359-368
Abstract
The total \(\pi\)-electron energy, as calculated within the Hückel tight-binding molecular orbital approximation, is a quantum-theoretical characteristic of conjugated molecules that has been conceived as early as in the 1930s. In 1978, a minor modification of the definition of total \(\pi\)-electron energy was put forward, that made this quantity interesting and attractive to mathematical investigations. The concept of graph energy, introduced in 1978, became an extensively studied graph-theoretical topic, with many hundreds of published papers. A great variety of graph energies is being considered in the current mathematical-chemistry and mathematical literature. Recently, some unexpected applications of these graph energies were discovered, in biology, medicine, and image processing.
We provide historic, bibliographic, and statistical data on the research on total \(\pi\)-electron energy and graph energies, and outline its present state of art. The goal of this survey is to provide, for the first time, an as-complete-as-possible list of various existing variants of graph energy, and thus help the readers to avoid getting lost in the jungle of references on this topic.
Boris Furtula, Ivan Gutman
Borderenergetic graphs of order 12
Iranian Journal of Mathematical Chemistry 8 (2017) 339-343
Abstract
A graph \(G\) of order \(n\) is said to be borderenergetic if its energy is equal to \(2n-2\) and if \(G\) differs from the complete graph \(K_n\). The first such graph was discovered in 2001, but their systematic study started only in 2015. Until now, the number of borderenergetic graphs of order \(n\) was determined for \(n \leq 11\). We now establish that there exist exactly 572 connected borderenergetic graphs of order 12.
Ivan Gutman, Kinkar C. Das, Boris Furtula, Emina Milovanović, Igor Milovanović
Generalizations of Szőkefalvi Nagy and Chebyshev inequalities with applications in spectral graph theory
Applied Mathematics and Computation 313 (2017) 235-244
Abstract
Two weighted inequalities for real non-negative sequences are proven. The first one represents a generalization of the Szőkefalvi-Nagy inequality for the variance, and the second a generalization of the discrete Chebyshev inequality for two real sequences. Then, the obtained inequalities are used to determine lower bounds for some degree-based topological indices of graphs.
Marija Antić, Boris Furtula, Slavko Radenković
Aromaticity of non-planar fully benzenoid hydrocarbons
The Journal of Physical Chemistry A 121 (2017) 3616-3626
Abstract
The Clar aromatic sextet theory can provide a qualitative description of the dominant modes of cyclic \(\pi\)-electron conjugation in benzenoid molecules and of the relative stability among a series of isomeric benzenoid systems. In a series of nonplanar fully benzenoid hydrocarbons, the predictions of the Clar theory were tested by means of several different theoretical approaches: topological resonance energy (\(TRE\)), energy effect (\(ef\)), harmonic oscillator model of aromaticity (\(HOMA\)) index, six center delocalization index (\(SCI\)), and nucleus-independent chemical shifts (\(NICS\)). To assess deviations from planarity in the examined molecules, four different planarity descriptors were employed. It was shown how the planarity indices can be used to quantify the effect of nonplanarity on the local and global aromaticity of the studied systems.
Ivan Gutman, Boris Furtula, Kinkar C. Das
Extended energy and its dependence on molecular structure
Canadian Journal of Chemistry 95 (2017) 526-529
Abstract
The extended energy (\(\mathcal{E}_{ex}\)) is a vertex degree based and spectrum-based molecular structure descriptor, shown to be well correlated with a variety of physicochemical molecular properties. We investigate the dependence of \(\mathcal{E}_{ex}\) on molecular structure and establish its basic characteristics. In particular, we show how \(\mathcal{E}_{ex}\) is related with the geometric–arithmetic (\(GA\)) topological index. Our main finding is that the difference between \(\mathcal{E}_{ex}\) and the total \(\pi\)-electron energy is linearly proportional to the difference between the number of edges and the \(GA\) index.
Bojana Borovićanin, Kinkar C. Das, Boris Furtula, Ivan Gutman
Bounds for Zagreb indices
MATCH Communications in Mathematical and in Computer Chemistry 78 (2017) 17-100
Abstract
Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). Let \(d_i\) be the degree of the vertex \(v_i \in V(G)\). The first and second Zagreb indices, \(M_1 = \sum_{v_i \in V(G)} d_i^2\) and \(M_2 = \sum_{v_iv_j \in E(G)} d_i\,d_j\) are the oldest and most thoroughly investigated vertex degree-based molecular structure descriptors. An unusually large number of lower and upper bounds for \(M_1\) and \(M_2\) have been established. We provide a survey of the most significant estimates of this kind, attempting to cover the existing literature up to the end of year 2016.
Igor Milovanović, Emina Milovanović, Ivan Gutman, Boris Furtula
Some inequalities for the forgotten topological index
International Journal of Applied Graph Theory 1 (2017) 1-15
Abstract
Let \(G= (V,E)\) be a simple connected graph with vertex set \(V=\{1,2,\ldots,n\}\) and edge set \(E=\{e_1, e_2,\ldots,e_m\}\). Let \(d_i\) be the degree of its vertex \(i\) and \(d(e_i)\) the degree of its edge \(e_i\). We consider the recently introduced degree–based graphinvariants: the forgotten index \(F = \sum_{i\in V} d_i^3\), the hyper-Zagreb index \(HM = \sum_{i\sim j} (d_i + d_j)^2\), and the reformulated first Zagreb index \(EM_1 = \sum_{e_i\in E} d(e_i)^2\). A number of lower and upper bounds for \(F\), \(HM\), and \(EM_1\) are established, and the equality cases determined.
Kinkar C. Das, Ivan Gutman, Boris Furtula
On spectral radius and energy of extended adjacency matrix of graphs
Applied Mathematics and Computation 296 (2017) 116-123
Abstract
Let \(G\) be a graph of order \(n\). For \(i + 1,2,\ldots,n\) let \(d_i\) be the degree of the vertex \(v_i\) of \(G\). The extended adjacency matrix \(\mathbf{A}_{ex}\) of \(G\) is defined so that its \((i,j)\)-entry is equal to \(\frac{1}{2}\left(\frac{d_i}{d_j} + \frac{d_j}{d_i}\right)\) if the vertices \(v_i\) and \(v_j\) are adjacent, and 0 otherwise, Yang et al. (1994). The spectral radius \(\eta_1\) and the energy \(\mathcal{E}_{ex}\) of the \(\mathbf{A}_{ex}\)-matrix are examined. Lower and upper bounds on \(\eta_1\) and \(\mathcal{E}_{ex}\) are obtained, and the respective extremal graphs characterized.
Ivan Gutman, Boris Furtula, Alexander Farrugia, Irene Sciriha
Constructing NSSD molecular graphs
Croatica Chemica Acta 89 (2016) 449-454
Abstract
A graph is said to be non-singular if it has no eigenvalue equal to zero; otherwise it is singular. Molecular graphs that are non-singular and also have the property that all subgraphs of them obtained by deleting a single vertex are themselves singular, known as NSSD graphs, are of importance in the theory of molecular \(žpi\)-electron conductors; NSSD = non-singular graph with a singular deck. Whereas all non-singular bipartite graphs (therefore, the molecular graphs of all closed-shell alternant conjugated hydrocarbons) are NSSD, the non-bipartite case is much more complicated. Only a limited number of non-bipartite molecular graphs have the NSSD property. Several methods for constructing such molecular graphs are presented.
Miloš Ivanović, Boban Stojanović, Višnja Simić, Ana Kaplarević Mališić, Vladimir Ranković, Boris Furtula, Stevan Mijailović
High performance computing in multi-scale modeling, graph science and meta-heuristic optimization
Journal of the Serbian Society for Computational Mechanics 10 (2016) 50-70
Abstract
One of the main activities within the Group for Scientific Computing at the Faculty of Science are methods for efficiently utilizing real parallel architectures, typically clusters of SMP nodes, shared-memory systems, and GPUs. Focus is on the design, development and implementation of parallel algorithms and data structures for fundamental scientific and engineering problems. Message Passing Interface (MPI) is an important paradigm that still poses interesting design and implementation problems, especially combined with other programming models, like CUDA. In addition to standard HPC (High Performance Computing) technology stack, the Group also utilizes computing stacks like Hadoop and Spark. In this paper we present a short review of the recent research of the Group, focused on large-scale applications in various research fields with references to original articles. The first part considers multi-scale muscle modeling in mixed MPI-CUDA environment. In our approach, a finite element macro model is coupled with the microscopic Huxley kinetics model. The original approach in scheduling tasks within multi-scale simulation ensures good load balance, leading to speed-up of over two orders of magnitude and high scalability. The second part considers application of HPC in graph science for the task of establishing the basic structural features of the minimum-\(ABC\) index trees. In order to analyze large amounts of data (all trees of certain order) we utilize grid computing services like storage and computing in order to reduce analysis time up to three orders of magnitude. The last part presents WoBinGO framework for solving optimization problems on HPC resources. It overcomes the shortcomings of earlier static pilot-job frameworks by providing elastic resource provisioning using adaptive allocation of jobs with limited lifetime. The obtained results show that despite WoBinGO’s adaptive and frugal allocation of computing resources, it provides significant speed-up when dealing with problems with computationally expensive evaluations, as found in hydro-informatics and market risk management.
Ivan Gutman, Boris Furtula, Kinkar C. Das
On some degree-and-distance-based graph invariants of trees
Applied Mathematics and Computation 289 (2016) 1-6
Abstract
Let \(G\) be a connected graph with vertex set \(V(G)\). For \(u, v \in V(G)\), \(d(v)\) and \(d(u,v)\) denote the degree of the vertex \(v\) and the distance between the vertices \(u\) and \(v\). A much studied degree–and–distance–based graph invariant is the degree distance, defined as \(DD = \sum_{\{u,v\}\subseteq V(G)} [d(u) + d(v)]d(u,v)\). A related such invariant (usually called "Gutman index") is \(ZZ = \sum_{\{u,v\}\subseteq V(G)} [d(u) \cdot d(v)]d(u,v)\). If \(G\) is a tree, then both \(DD\) and \(ZZ\) are linearly related with the Wiener index \(W = \sum_{\{u,v\}\subseteq V(G)} d(u,v)\). We examine the difference \(DD - ZZ\) for trees and establish a number of regularities.
Boris Furtula, Ivan Gutman, Kinkar C. Das
On atom-bond connectivity molecule structure descriptors
Journal of the Serbian Chemical Society 81 (2016) 271-276
Abstract
The atom-bond connectivity index (\(ABC\)) is a degree-based molecular structure descriptor with well-documented chemical applications. In 2010 a distance-based new variant of this index (\(ABC_{GG}\)) has been proposed. Until now, the relation between \(ABC\) and \(ABC_{GG}\) has not been analyzed. In this paper, we establish the basic characteristics of this relation. In particular, \(ABC\) and \(ABC_{GG}\) are not correlated and both cases \(ABC \gt ABC_{GG}\) and \(ABC \lt ABC_{GG}\) may occur in the case of (structurally similar) molecules. However, in the case of benzenoid hydrocarbons, \(ABC\) always exceeds \(ABC_{GG}\).
Bojana Borovićanin, Boris Furtula
On extremal Zagreb indices of trees with given domination number
Applied Mathematics and Computation 279 (2016) 208-218
Abstract
First (\(M_1\)) and second (\(M_2\)) Zagreb indices are graph invariants that originate from chemical researches on total π-electron energy of conjugated molecules. There is a legion of articles dealing with these two indices. This paper presents upper bounds on Zagreb indices of trees in terms of domination number. These are strict bounds, and extremal trees are characterized. In addition, a lower bound for the first Zagreb index of trees with a given domination number is determined and the extremal trees are characterized as well. Finally, using previously known upper bound for Harary index (\(H\)) in terms of \(M_1\) and \(M_2\), a unique tree with given domination number that maximizes H is characterized.
Boris Furtula, Ivan Gutman, Vladimir Katanić
Three-center Harary index and its applications
Iranian Journal of Mathematical Chemistry 7 (2016) 61-68
Abstract
The Harary index \(H\) can be viewed as a molecular structure descriptor composed of increments representing interactions between pairs of atoms, such that their magnitude decreases with the increasing distance between the respective two atoms. A generalization of the Harary index, denoted by \(H_k\), is achieved by employing the Steiner-type distance between \(k\)-tuples of atoms. We show that the linear combination \(H + \lambda\,H_3\) is significantly better correlated with a variety of physico-chemical properties of alkanes than \(H\) itself.
Ivan Gutman, Boris Furtula, Emir Zogić, Edin Glogić
Resolvent energy of graphs
MATCH Communications in Mathematical and in Computer Chemistry 75 (2016) 279-290
Abstract
The resolvent energy of a graph \(G\) of order \(n\) is defined as \(ER = \sum_{i=1}^n (n - \lambda_i)^{-1}\), where \(\lambda_1, \lambda_2, \ldots\lambda_n\) are the eigenvalues of \(G\). We establish a number of properties of \(ER\). In particular, we establish lower and upper bounds for \(ER\) and characterize the trees, unicyclic, and bicyclic graphs with smallest and greatest \(ER\).
Boris Furtula
Atom–bond connectivity index versus Graovac–Ghorbani analog
MATCH Communications in Mathematical and in Computer Chemistry 75 (2016) 233-242
Abstract
The atom-bond connectivity index was introduced almost twenty years ago as an improvement of the well-known Randić index. Its mathematical properties and theory are well established and chemical usefulness confirmed in various research projects. On the other hand, recently introduced quantity that was named as second \(ABC\) index resembles to the original-one, but it is not validated anywhere as a molecular descriptor. Also its mathematical properties are examined to a limited extent. This paper is devoted to the comparison of these two topological invariants.
Boris Furtula, Ivan Gutman, Žana Kovijanić Vukićević, Giorgi Lekishvili, Goran Popivoda
On an old/new degree-based topological index
Bulletin de l'Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 40 (2015) 19-31
Abstract
Let \(G\) be a graph with vertex set \(V(G)\) and let \(d(x)\) be the degree of the vertex \(x \in V(G)\). The graph invariant \(F = \sum_{x\in V(G)} d(x)^3\) played some role in a paper published in 1972, but has not attracted any attention until quite recently. In 2014 an unexpected chemical application of the \(F\)-index was discovered, which motivated us to establish its basic mathematical properties. Results obtained along these lines are presented.
Ivan Gutman, Boris Furtula, Xiaodan Chen, Jianguo Qian
Resolvent Estrada index – computational and mathematical studies
MATCH Communications in Mathematical and in Computer Chemistry 74 (2015) 431-440
Abstract
The resolvent Estrada index of a (non-complete) graph \(G\) of order \(n\) is defined as \(EE_r =\sum_{i=1}^n(1-\lambda_i/(n-1))^{-1}\), where \(\lambda_1, \lambda_2, \lambda_n\) are the eigenvalues of \(G\). Combining computational and mathematical approaches, we establish a number of properties of \(EE_r\). In particular, any tree has smaller \(EE_r\)-value than any unicyclic graph of the same order, and any unicyclic graph has smaller \(EE_r\)-value than any tricyclic graph of the same order. The trees, unicyclic, bicyclic, and tricyclic graphs with smallest and greatest \(EE_r\) are determined.
Shicai Gong, Xueliang Li, Guanghui Xu, Ivan Gutman, Boris Furtula
Borderenergetic graphs
MATCH Communications in Mathematical and in Computer Chemistry 74 (2015) 321-332
Abstract
The energy \(\mathcal{E}(G)\) of a graph \(G\) is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. A graph \(G\) of order \(n\) is said to be borderenergetic if its energy equals the energy of the complete graph \(K_n\), i.e., if \(\mathcal{E}(G) = 2(n - 1)\). We first show by examples that there exist connected borderenergetic graphs, different from the complete graph \(K_n\). The smallest such graph is of order 7. We then show that for each integer \(n\), \(n \geq 7\), there exists borderenergetic graphs of order \(n\), different from \(K_n\), and describe the construction of some of these graphs.
Ivan Gutman, Boris Furtula, Xueliang Li
Multicenter Wiener indices and their applications
Journal of the Serbian Chemical Society 80 (2015) 1009-1017
Abstract
The Wiener index \(W\) can be viewed as a molecular structure descriptor composed of increments representing interactions between pairs of atoms. A generalization of the \(W\) are the Steiner-Wiener indices \(W_k, k=3,4,\ldots\). In the quantity \(W_k\), interactions between \(k\)-tuples of atoms play role, based on the concept of Steiner distance. It is shown that the term \(W + \lambda\,W_k\) provides an approximation for the boiling points of alkanes better than \(W\) itself. The best such approximation is obtained for \(k = 7\).
Ivan Gutman, Boris Furtula, Žana Kovijanić Vukićević, Goran Popivoda
On Zagreb indices and coindices
MATCH Communications in Mathematical and in Computer Chemistry 74 (2015) 5-16
Abstract
A complete set of relations is established between the first and second Zagreb index and coindex of a graph and of its complements. Formulas for the first Zagreb index of several derived graphs are also obtained. A remarkable result is that the first Zagreb coindices of a graph and of its complement are always equal.
Boris Furtula
Assessing \(\pi\)-electron contents of rings in polycyclic aromatic compounds
Current Organic Chemistry 19 (2015) 331-347
Abstract
Polycyclic aromatic compounds (\(PAC\)) are a huge class of organic molecules that can be found everywhere. They called attention of scientists by their interesting and sometimes peculiar physical and chemical properties. That is why, there are a number of research articles and books in which their synthesis, properties, reactions, and theoretical investigations have been communicated. In this survey will be outlined several simple methods for assessing the partition of π-electrons in rings of some classes of \(PAC\).
Boris Furtula, Ivan Gutman
A forgotten topological index
Journal of Mathematical Chemistry 53 (2015) 1184-1190
Abstract
In 1972, within a study of the structure-dependency of total \(\pi\)-electron energy (\(\mathcal{E}\)), it was shown that \(\mathcal{E}\) depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. In the same paper, also the sum of cubes of degrees of vertices of the molecular graph was shown to influence \(\mathcal{E}\), but this topological index was never again investigated and was left to oblivion. We now establish a few basic properties of this "forgotten topological index" and show that it can significantly enhance the physico-chemical applicability of the first Zagreb index.
Ivan Gutman, Boris Furtula, Xiaodan Chen, Jianguo Qian
Graphs with smallest resolvent Estrada indices
MATCH Communications in Mathematical and in Computer Chemistry 73 (2015) 267-270
Abstract
The graphs and trees with smallest resolvent Estrada indices (\(EE_r\)) are characterized. The connected graph of order n with smallest \(EE_r\)-value is the \(n\)-vertex path. The second-smallest such graph is the \((n-1)\)-vertex path with a pendent vertex attached at position 2. The tree with third-smallest \(EE_r\) is the \((n-1)\)-vertex path with a pendent vertex attached at position 3, conjectured to be also the connected graph with third-smallest \(EE_r\). Based on a computer-aided search, we established the structure of a few more trees with smallest \(EE_r\).
Boris Furtula, Ivan Gutman, Süleyman Ediz
On difference of Zagreb indices
Discrete Applied Mathematics 178 (2014) 83-88
Abstract
The classical first and second Zagreb indices of a graph \(G\) are defined as \(M_1 = \sum_v d_v^2\) and \(M_2 = \sum_{uv} d_u\,d_v\), where \(d_v\) is the degree of the vertex \(v\) of \(G\). So far, the difference of \(M_1\) and \(M_2\) has not been studied. We show that this difference is closely related to the vertex-degree-based invariant \(RM_2 = \sum_{uv} (d_u − 1)(d_v − 1)\), and determine a few basic properties of \(RM_2\).
Boris Furtula, Giorgi Lekishvili, Ivan Gutman
A graph theoretical approach to cis/trans isomerism
Journal of the Serbian Chemical Society 79 (2014) 805-813
Abstract
A simple graph-theory-based model is put forward, by means of which it is possible to express the energy difference between geometrically non-equivalent forms of a conjugated polyene. This is achieved by modifying the adjacency matrix of the molecular graph, and including into it information on cis/trans constellations. The total \(\pi\)-electron energy thus calculated is in excellent agreement with the enthalpies of the underlying isomers and conformers.
Ivan Gutman, Boris Furtula, Clive Elphick
Three new/old vertex-degree-based topological indices
MATCH Communications in Mathematical and in Computer Chemistry 72 (2014) 617-632
Abstract
Three vertex-degree-based graph invariants are presented, that earlier have been considered in the chemical and/or mathematical literature, but that evaded the attention of most mathematical chemists. These are the reciprocal Randić index (\(RR\)), the reduced second Zagreb index \(RM_2\), and the reduced reciprocal Randić index (\(RRR\)). If \(d_1, d_2,\ldots, d_n\) are the degrees of the vertices of the graph \(G = (V,E)\), then
\[
RR = \sum_{ij\in E} \sqrt{d_i\,d_j} \quad RM_2 = \sum_{ij\in E} (d_i - 1)(d_j - 1) \quad RRR = \sum_{ij\in E} \sqrt{(d_i - 1)(d_j - 1)} \ .
\]
We outline the literature sources of these topological indices, their main mathematical properties, and establish their correlating abilities w.r.t. characteristic physico-chemical properties of alkanes.
Ivan Gutman, Boris Furtula, Vladimir Katanić
New bounds and approximations for total \(\pi\)-electron energy - a critical test
Kragujevac Journal of Science 36 (2014) 79-86
Abstract
The quality and correlating ability of some recently deduced bounds and approximate formulas for total \(\pi\)-electron energy are tested on a sample of 106 Kekuléan benzenoid hydrocarbons. It was found that not a single new approximate formula and not a single new bound is better than those from the 1970s.
Ivan Gutman, Boris Furtula, Şerife Burcu Bozkurt
On Randić energy
Linear Algebra and its Applications 442 (2014) 50-57
Abstract
The Randić matrix \(\mathbf{R} = (r_{ij})\) of a graph \(G\) whose vertex \(v_i\) has degree \(d_i\) is defined by \(r_{ij} = 1/\sqrt{d_i,d_j}\) if the vertices \(v_i\) and \(v_j\) are adjacent and \(r_{ij} = 0\) otherwise. The Randić energy \(RE\) is the sum of absolute values of the eigenvalues of \(\mathbf{R}\). \(RE\) coincides with the normalized Laplacian energy and the normalized signless-Laplacian energy. Several properties of \(\mathbf{R}\) and \(RE\) are determined, including characterization of graphs with minimal \(RE\). The structure of the graphs with maximal \(RE\) is conjectured.
Kexiang Xu, Muhuo Liu, Kinkar C. Das, Ivan Gutman, Boris Furtula
A survey on graphs extremal with respect to distance-based topological indices
MATCH Communications in Mathematical and in Computer Chemistry 71 (2014) 461-508
Abstract
This survey outlines results on graphs extremal with respect to distance-based indices, with emphasis on the Wiener index, hyper-Wiener index, Harary index, Wiener polarity index, reciprocal complementary Wiener index, and terminal Wiener index.
Ivan Gutman, Boris Furtula, Jelena Tošović, Mohamed Essalih, Mohamed El Marraki
On terminal Wiener indices of kenograms and plerograms
Iranian Journal of Mathematical Chemistry 4 (2013) 77-89
Abstract
Whereas there is an exact linear relation between the Wiener indices of kenograms and plerograms of isomeric alkanes, the respective terminal Wiener indices exhibit a completely different behavior: Correlation between terminal Wiener indices of kenograms and plerograms is absent, but other regularities can be envisaged. In this article, we analyze the basic properties of terminal Wiener indices of kenograms and plerograms.
Boris Furtula, Ivan Gutman, Matthias Dehmer
On structure-sensitivity of degree-based topological indices
Applied Mathematics and Computation 219 (2013) 8973-8978
Abstract
One of the general requirements for any topological index \(TI\) is that similar molecules have near-lying \(TI\)-values, which is referred to as "smoothness". Curiously, however, smoothness of topological indices was until now never examined and never quantified. We now propose a pertinent mathematical model for this property, and undertake a comparative study of the structure-sensitivity of 12 degree-based topological indices by using trees.
Ivan Gutman, Mohamed Essalih, Mohamed El Marraki, Boris Furtula
Why plerograms are not used in chemical graph theory? The case of terminal-Wiener index
Chemical Physics Letters 568-569 (2013) 195-197
Abstract
The reasons why plerogram-type molecular graphs are almost never used in contemporary chemical graph theory are analyzed. As one more argument against the usage of plerograms, we show that the terminal-Wiener index of the plerogram of an alkane is in an (exact) linear manner related to the ordinary Wiener index, thus having the precisely same structure dependency as the ordinary Wiener index.
Boris Furtula
Odd-vertex-degree trees maximizing Wiener index
Kragujevac Journal of Mathematics 37 (2013) 129-134
Abstract
Wiener index is the oldest and one among the most investigated topological indices in mathematical chemistry and in related disciplines. Recently, Wiener index of odd-vertex-degree (\(T^{odd}\)) trees has been investigated. In this paper, trees with second, third, ..., seventeenth maximal Wiener index are characterized.
Boris Furtula, Ivan Gutman
Comparing energy and Randić energy
Macedonian Journal of Chemistry and Chemical Engineering 32 (2013) 117-123
Abstract
The recently conceived Randić energy (\(RE\)) is examined, and its relation to the (earlier much studied) total \(\pi\)-electron energy (\(E\)) is investigated. Within classes of molecular graphs, there exists a relatively good (increasing) linear correlation between \(RE\) and \(E\). However, several significant differences between the structure-dependencies of \(RE\) and \(E\) have been discovered, the most striking of which is their dependence on the number \(m\) of edges of the underlying graph. Whereas, with increasing \(m\), the average value of \(E\) increases, reaches a maximum and then decreases, the average value of \(RE\) monotonically decreases. The structure of the connected graph with a fixed number of vertices and maximal \(RE\) value was established.
Ivan Gutman, Boris Furtula, Mohammad B. Ahmadi, Seyyed A. Hosseini, Salehi P. Nowbandegani, Maryam Zarrinderakht
The ABC index conundrum
FILOMAT 27 (2013) 1075-1083
Abstract
The atom-bond connectivity (\(ABC\)) index of a graph \(G\) is defined as the sum over all pairs of adjacent vertices \(u\), \(v\), of the terms \(\sqrt{[d(u) + d(v)-2]/[d(u)\,d(v)]}\), where \(d(v)\) denotes the degree of the vertex \(v\) of the graph \(G\). Whereas the finding of the graphs with the greatest \(ABC\)-value is an easy task, the characterization of the graphs with smallest \(ABC\)-value, in spite of numerous attempts, is still an open problem. What only is known is that the connected graph with minimal \(ABC\) index must be a tree, and some structural features of such trees have been determined. Several conjectures on the structure of the minimal-\(ABC\) trees, were disproved by counterexamples. In this review we present the state of art of the search for minimal-\(ABC\) trees, and provide a complete bibliography on \(ABC\) index.
Boris Furtula, Ivan Gutman, Hong Lin
More trees with all degrees odd having extremal Wiener index
MATCH Communications in Mathematical and in Computer Chemistry 70 (2013) 293-296
Abstract
Continuing the preceding work [MATCH Commun. Math. Comput. Chem. 70 (2013) 000-000], we determine the first few trees whose all degrees are odd, having smallest and greatest Wiener index.
Boris Furtula, Ivan Gutman, Miloš Ivanović, Damir Vukičević
Computer search for trees with minimal \(ABC\) index
Applied Mathematics and Computation 219 (2012) 767-772
Abstract
The \(ABC\) index is a degree-based molecular structure descriptor, that found chemical applications. Finding the connected graph(s) of a given order whose \(ABC\) index is minimal is a hitherto unsolved problem, but it is known that these must be trees. In this paper, by combining mathematical arguments and computer-based modeling we establish the basic structural features of the minimum-\(ABC\) trees.
Matthias Dehmer, Martin Grabner, Boris Furtula
Structural discrimination of networks by using distance, degree and eigenvalue-based measures
PloS ONE 7 (2012) e38564-(7)15
Abstract
In chemistry and computational biology, structural graph descriptors have been proven essential for characterizing the structure of chemical and biological networks. It has also been demonstrated that they are useful to derive empirical models for structure-oriented drug design. However, from a more general (complex network-oriented) point of view, investigating mathematical properties of structural descriptors, such as their uniqueness and structural interpretation, is also important for an in-depth understanding of the underlying methods. In this paper, we emphasize the evaluation of the uniqueness of distance, degree and eigenvalue-based measures. Among these are measures that have been recently investigated extensively. We report numerical results using chemical and exhaustively generated graphs and also investigate correlations between the measures.
Ivan Gutman, Boris Furtula
Trees with smallest atom–bond connectivity index
MATCH Communications in Mathematical and in Computer Chemistry 68 (2012) 131-136
Abstract
The structure of trees with a single high-degree vertex and smallest \(ABC\) index is determined.
Ivan Gutman, Boris Furtula
Vertex-degree-based molecular structure descriptors of benzenoid systems and phenylenes
Journal of the Serbian Chemical Society 77 (2012) 1031-1036
Abstract
Several recently published papers report expressions for various vertex-degree-based molecular structure descriptors of benzenoid systems and phenylenes. We deduce here the general expression for these descriptors, and show that a simple and generally valid relation exists between such structure descriptors of phenylenes and their hexagonal squeezes.
Hossein Shabani, Alireza Ashrafi, Ivan Gutman, Boris Furtula
On extensions of Wiener index
MATCH Communications in Mathematical and in Computer Chemistry 69 (2013) 589-596
Abstract
The \(n\)-th order Wiener index of a molecular graph \(G\) was put forward by E. Estrada et al. [New J. Chem. 22 (1998) 819] as \(n^W = H^{(n)} (G,x)|_{x=1}\) where \(H(G,x)\) is the Hosoya polynomial. Recently F. M. Brückler et al. [Chem. Phys. Lett. 503 (2011) 336] considered a related graph invariant, \(W^{(n)} = (1/n!)d^n (x^{n-1} H(G,x))/dx^n|_{x=1}\). For \(n=1\), both \(n^W\) and \(W^{(n)}\) reduce to the ordinary Wiener index. The aim of this paper is to obtain closed formulas for these two extensions of the Wiener index. It is proved that \(W^{(n)} =(1/n!)\sum_{k=1}^n c(n,k)W_k\) and \(n^W = \sum_{k=1}^n s(n,k)W_k\), where \(c(n,k)\), \(s(n,k)\), and \(W_k\) stand for the unsigned Stirling number of the first kind, Stirling number of the first kind, and the \(k\)-th distance moment of \(G\), respectively.
Ivan Gutman, Boris Furtula, Eric O. D. Andriantiana, Mila Cvetić
More trees with large energy and small size
MATCH Communications in Mathematical and in Computer Chemistry 68 (2012) 697-702
Abstract
In a previous paper [E. O. D. Andriantiana, MATCH Commun. Math Comput. Chem. 68 (2012) 000-000] trees with a fixed number \(n\) of vertices were ordered according to their energy, and a large number of trees with greatest energy were characterized. These results, however, hold only if \(n\) is large enough. We now analyze the energy-ordering of trees for small values of \(n\) (up to 100) and establish the first few greatest-energy species. The results obtained for small values of \(n\) significantly differ from those valid for large values of \(n\).
Kinkar C. Das, Ivan Gutman, Boris Furtula
On atom-bond connectivity index
FILOMAT 26 (2012) 733-738
Abstract
The atom-bond connectivity index (\(ABC\)) is a vertex-degree based graph invariant, put forward in the 1990s, having applications in chemistry. Let \(G = (V,E)\) be a graph, \(d_i\) the degree of its vertex \(i\), and \(ij\) the edge connecting the vertices \(i\) and \(j\). Then \(ABC = \sum_{ij\in\mathbf{E}} \sqrt{(d_i + d_j - 2) / (d_i\,d_j)}\). Upper bounds and Nordhaus-Gaddum type results for \(ABC\) are established.
Ivan Gutman, Boris Furtula, Miloš Ivanović
Notes on trees with minimal atom-bond connectivity index
MATCH Communications in Mathematical and in Computer Chemistry 67 (2012) 467-482
Abstract
If \(G = (\mathbf{V, E})\) is a molecular graph, and \(d(u)\) is the degree of its vertex \(u\), then the atom-bond connectivity index of \(G\) is \(ABC = \sum_{uv\in\mathbf{E}} \sqrt{[d(u) + d(v) - 2] / [d(u)\,d(v)]}\). This molecular structure descriptor, introduced by Estrada et al. in the late 1990s, found recently interesting applications in the study of the thermodynamic stability of acyclic saturated hydrocarbons, and the strain energy of their cyclic congeners. In connection with this, one needs to know which trees have extremal \(ABC\)-values. Whereas it is easy to demoastrate that the star has maximal \(ABC\), characterizing the trees with minimal \(ABC\) appears to be a much more difficult task. In this paper we determine a few structural features of the trees with minimal \(ABC\), which brings us a step closer to the complete solution of the problem.
Kinkar C. Das, Ivan Gutman, Boris Furtula
On the first geometric-arithmetic index of graphs
Discrete Applied Mathematics 159 (2011) 2030-2037
Abstract
Let \(G\) be a simple connected graph and \(d_i\) be the degree of its \(i\)th vertex. In a recent paper [D. Vukičević, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009) 1369–1376] the “first geometric–arithmetic index” of a graph \(G\) was defined as
\[
GA_1 = \sum \frac{\sqrt{d_i\,d_j}}{(d_i + d_j)/2}
\]
with summation going over all pairs of adjacent vertices. We obtain lower and upper bounds on \(GA_1\) and characterize graphs for which these bounds are best possible. Moreover, we discuss the effect on \(GA_1\) of inserting an edge into a graph.
Kinkar C. Das, Ivan Gutman, Boris Furtula
On atom-bond connectivity index
Chemical Physics Letters 511 (2011) 452-454
Abstract
The atom-bond connectivity index (\(ABC\)) is a molecular structure descriptor that recently found a remarkable application in rationalizing the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. Let \(G\) be a molecular graph, \(d_i\) the degree of its vertex \(i\), and \(ij\) the edge connecting the vertices \(i\) and \(j\). Then \(ABC = \sum_{ij} \sqrt{(d_i + d_j - 2) / (d_i\,d_j)}\). Several properties of \(ABC\) are established. In particular, if a new edge is inserted into \(G\), then \(ABC\) necessarily increases. By means of this result we characterize the graphs extremal w.r.t. \(ABC\).
Boris Furtula, Ivan Gutman
Relation between second and third geometric–arithmetic indices of trees
Journal of Chemometrics 25 (2011) 87-91
Abstract
The geometric–arithmetic indices (\(GA\)) are a recently introduced class of molecular structure descriptors found to be useful tools in QSPR/QSAR researches. We now establish a peculiar relation between the second (\(GA_2\)) and the third (\(GA_3\)) geometric–arithmetic indices of trees and chemical trees: for trees with a fixed number of vertices (\(n\)) and pendent vertices (\(\nu\)), \(GA_2\) and \(GA_3\) are almost exactly linearly correlated. For various values of \(\nu\), the \(GA_3/GA_2\) lines are parallel, and their distance is proportional to \(\nu\). These findings are rationalized by deducing lower and upper bounds for \(GA_3\) that are increasing linear functions of \(GA_2\) and decreasing linear functions of \(\nu\).
Ivan Gutman, Boris Furtula, Alexandru T. Balaban
Effect of benzocyclobutadieno-annelation on cyclic conjugation in fluoranthene congeners
Journal of the Serbian Chemical Society 76 (2011) 733-741
Abstract
Earlier studies revealed that benzo-annelation has a peculiar effect on the intensity of cyclic conjugation in the five-membered ring of fluoranthene congeners. Now, the analogous effect of benzocyclobutadieno-annelation was examined and it was found that it is opposite to the effect of benzo-annelation: A benzocyclobutadiene fragment in angular (resp. linear) position with regard to the five-membered ring, decreases (resp. increases) the intensity of cyclic conjugation in the five-membered ring.
Ivan Gutman, Boris Furtula
Estimating the second and third geometric-arithmetic indices
Hacettepe Journal of Mathematics and Statistics 40 (2011) 69-76
Abstract
Arithmetic-geometric indices are graph invariants defined as the sum of terms \(\sqrt{Q_uQ_v}/[(Q_u + Q_v)/2]\) over all edges \(uv\) of the graph, where \(Q_u\) is some quantity associated with the vertex \(u\). If \(Q_u\) is the number of vertices (resp. edges) lying closer to \(u\) than to \(v\), then one speaks of the second (resp. third) geometric–arithmetic index, \(GA_2\) and \(GA_3\). We obtain inequalities between \(GA_2\) and \(GA_3\) for trees, revealing that the main parameters determining their relation are the number of vertices and the number of pendent vertices.
Damir Vukičević, Ivan Gutman, Boris Furtula, Vesna Andova, Darko Dimitrov
Some observations on comparing Zagreb indices
MATCH Communications in Mathematical and in Computer Chemistry 66 (2011) 627-645
Abstract
Let \(G\) be a simple graph possessing \(n\) vertices and \(m\) edges. Let \(d_i\) be the degree of the \(i\)-th vertex of \(G, i=1,\ldots,n\). The first Zagreb index \(M_1\) is the sum of \(d_i^2\) over all vertices of \(G\). The second Zagreb index \(M_2\) is the sum \(d_i\,d_j\) over pairs of adjacent vertices of \(G\). We search for a graph for which \(M_1/n = M_2/m\), and show how many ways such graphs can be constructed. In addition, we find examples of graphs for which \(M_1/n > M_2/m\) which are counterexamples to the earlier conjectured inequality \(M_1/n \leq M_2/m\).
Tomislav Došlić, Boris Furtula, Ante Graovac, Ivan Gutman, Sirous Moradi, Zahra Yarahmadi
On vertex-degree-based molecular structure descriptors
MATCH Communications in Mathematical and in Computer Chemistry 66 (2011) 613-626
Abstract
If \(G=(\mathbf{V} \ \mathbf{E})\) is a molecular graph and \(d_u\) is the degree of vertex \(u\), then the first and second Zagreb indices are \(\sum_{u\in \mathbf{V}} d_u^2\) and \(\sum_{uv\in \mathbf{E}} d_u\,d_v\),respectively. These molecular structure descriptors, introduced in the 1970s, have been much studied. Yet, a number of their properties that seem to have evaded attention so far are established in this work for the first time. Also a more recent degree-based descriptor, the geometric-arithmetic index, equal to \(\sum_{uv\in\mathbf{E}} \sqrt{d_u\,d_v}/[(d_u + d_v)/2]\), is analyzed. It is demonstrated that instead of the ratio of geometric and arithmetic means, almost any other means could be used.
Kinkar C. Das, Ivan Gutman, Boris Furtula
Survey on geometric-arithmetic indices of graphs
MATCH Communications in Mathematical and in Computer Chemistry 65 (2011) 595-644
Abstract
The concept of geometric-arithmetic indices (\(GA\)) was introduced in chemical graph theory very recently. In spite of this, several papers have already appeared dealing with these indices. The main goal of this survey is to collect all hitherto obtained results on \(GA\) indices (both chemical and mathematical).
Kinkar C. Das, Ivan Gutman, Boris Furtula
On third geometric-arithmetic index of graphs
Iranian Journal of Mathematical Chemistry 1 (2010) 29-36
Abstract
Continuing the work K. C. Das, I. Gutman, B. Furtula, On second geometric-arithmetic index of graphs, Iran. J. Math Chem. 1(2) (2010) 17-28, in this paper we present lower and upper bounds on the third geometric-arithmetic index \(GA_3\) and characterize the extremal graphs. Moreover, we give Nordhaus-Gaddum-type result for \(GA_3\).
Kinkar C. Das, Ivan Gutman, Boris Furtula
On second geometric-arithmetic index of graphs
Iranian Journal of Mathematical Chemistry 1 (2010) 17-28
Abstract
The concept of geometric-arithmetic indices (\(GA\)) was put forward in chemical graph theory very recently. In spite of this, several works have already appeared dealing with these indices. In this paper we present lower and upper bounds on the second geometric-arithmetic index (\(GA_2\)) and characterize the extremal graphs. Moreover, we establish Nordhaus-Gaddum-type results for \(GA_2\).
Boris Furtula, Ante Graovac, Damir Vukičević
Augmented Zagreb index
Journal of Mathematical Chemistry 48 (2010) 370-380
Abstract
Inspired by recent work on the atom-bond connectivity (\(ABC\)) index we propose here a new topological index, augmented Zagreb index (\(AZI\)). The tight upper and lower bounds for chemical trees are obtained. Moreover, it has been shown that among all trees the star has the minimum \(AZI\) value. Characterizing trees with maximal augmented Zagreb index remains an open problem for future research.
Boris Furtula, Ivan Gutman, Svetlana Jeremić, Slavko Radenković
Effect of a ring on cyclic conjugation in another ring: Applications to acenaphthylene-type polycyclic conjugated molecules
Journal of the Serbian Chemical Society 75 (2010) 83-90
Abstract
In a recent work, a method was developed for assessing the influence \(ief(G,Z_0|Z_1)\) of a ring \(Z_1\) on the energy effect of another ring \(Z_0\) in a polycyclic conjugated molecule \(G\). Herein, a report is given of detailed numerical investigations of \(ief(G,Z_0|Z_1)\) aimed at the elucidation of the influence of various six-membered rings on the intensity of cyclic conjugation in the fivemembered ring of acenaphthylene-type molecules. The earlier discovered regularities for cyclic conjugation in acenaphthylene-type molecules (in particular, the PCP rule and the linear rule) could thus not only be rationalized, but also a number of hitherto concealed regularities could be envisaged.
Gholamhossein Fath-Tabar, Boris Furtula, Ivan Gutman
A new geometric-arithmetic index
Journal of Mathematical Chemistry 47 (2010) 477-486
Abstract
A new molecular-structure descriptor \(GA_2\), belonging to the class of geometric–arithmetic indices, is considered. It is closely related to the Szeged and vertex \(PI\) indices. The main properties of \(GA_2\) are established, including lower and upper bounds. The trees with minimum and maximum \(GA_2\) are characterized.
Ivan Gutman, Bo Zhou, Boris Furtula
The Laplacian-energy like invariant is an energy like invariant
MATCH Communications in Mathematical and in Computer Chemistry 64 (2010) 85-96
Abstract
Short time ago Liu and Liu [MATCH Commun. Math. Comput. Chem. 59 (2008) 355-372] put forward a so-called Laplacian-energy like invariant (\(LEL\)), defined as the sum of the square roots of the Laplacian eigenvalues. From its name, one could get the impression that the properties of \(LEL\) are similar to those of the Laplacian energy \(LE\). However, already the inventors of \(LEL\) realized that \(LEL\) resembles much more the ordinary graph energy (\(E\)) than \(LE\). We now provide further arguments supporting this conclusion. In particular, numerous earlier obtained bounds and approximations for \(E\) can be simply "translated" into bounds and approximations for \(LEL\).
Gilles Caporossi, Emma Chasset, Boris Furtula
Some conjectures and properties on distance energy
Les Cahiers du GERAD G-2009-64 (2009) 1-7
Abstract
The distance energy of a graph \(G\) is defined as \(E_D(G) = \sum |\mu_i|\), where \(\mu_i\) is the \(i^{th}\) eigenvalue of the distance matrix of \(G\). In this paper, we express the distance spectra and distance energy of complete split graphs and graphs composed of two cliques joined by a matching. We also give some spectral properties of complete multipartite graphs. Finally, we identify structural and numerical conjectures on \(E_D\) for graphs with number of vertices \(n\) and number of edges \(m\) are fixed.
Bo Zhou, Ivan Gutman, Boris Furtula, Zhibin Du
On two types of geometric-arithmetic index
Chemical Physics Letters 482 (2009) 153-155
Abstract
Recently a class of so-called "geometric–arithmetic" topological indices (\(GA\)) was put forward, defined as the sum over all edges (\(uv\)) of a (molecular) graph \(G\), of terms \(\sqrt{Q_uQ_v}/\frac{1}{2}(Q_u + Q_v)\), where \(Q_u\) is some quantity associated with the vertex \(u\) of \(G\). One variant of \(GA\) was obtained for \(Q_u\) is the number of vertices of \(G\), lying closer to vertex \(u\) than to vertex \(v\). We obtain bounds for this \(GA\)-index, and also put forward an analogous one, for which \(Q_u\) is the number of edges of \(G\), lying closer to vertex \(u\) than to vertex \(v\).
Boris Furtula, Ante Graovac, Damir Vukičević
Atom-bond connectivity index of trees
Discrete Applied Mathematics 157 (2009) 2828-2835
Abstract
The recently introduced atom–bond connectivity (\(ABC\)) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Here, mathematical properties of the \(ABC\) index of trees are studied. Chemical trees with extremal \(ABC\) values are found. In addition, it has been proven that, among all trees, the star tree, Sn, has the maximal \(ABC\) value.
Damir Vukičević, Boris Furtula
Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges
Journal of Mathematical Chemistry 46 (2009) 1369-1376
Abstract
Research on the topological indices based on end-vertex degrees of edges has been intensively rising recently. Randić index, one of the best-known topological indices in chemical graph theory, is belonging to this class of indices. In this paper, we introduce a novel topological index based on the end-vertex degrees of edges and its basic features are presented here. We named it as geometrical-arithmetic index (\(GA\)).
Ivan Gutman, Boris Furtula, Miroslav Petrović
Terminal Wiener index
Journal of Mathematical Chemistry 46 (2009) 522-531
Abstract
Motivated by some recent research on the terminal (reduced) distance matrix, we consider the terminal Wiener index (\(TW\)) of trees, equal to the sum of distances between all pairs of pendent vertices. A simple formula for computing \(TW\) is obtained and the trees with minimum and maximum \(TW\) are characterized.
Olga Miljković, Boris Furtula, Slavko Radenković, Ivan Gutman
Equienergetic and almost-equienergetic trees
MATCH Communications in Mathematical and in Computer Chemistry 61 (2009) 451-461
Abstract
The energy \(E(G)\) of a graph \(G\) is equal to the sum of the absolute values of the eigenvalues of \(G\). Two graphs \(G_a\) and \(G_b\) are said to be equienergetic if \(E(G_a) = E(G_b)\). Numerous families of non-cospectral eqnienergetic graphs have been reported so far. However, until now it was not noticed that there exist pairs of graphs whose energies differ insignificantly. We refer to such graphs as almost-equienergetic. A detailed study of almost-equienergetic trees is provided.
Boris Furtula, Ivan Gutman
Energy and Estrada index of phenylenes
Indian Journal of Chemistry 47A (2008) 220-224
Abstract
In the theory of polycyclic conjugated molecules, several remarkable results are known relating the properties of a phenylene with the analogous properties of a benzenoid molecule that in a natural way is associated with \(PH\), called the hexagonal squeeze (\(HS\)) (for details see Fig. 1 or ref. 2). In the present work, the relationships between the energy, \(E\), and the Estrada index, \(EE\), of phenylenes and their hexagonal squeezes are examined. Within sets of isomers, a good linear correlation exists between \(E\)(phenylene) and \(E\)(hexagonal squeeze), as well as between \(EE\)(phenylene) and \(EE\)(hexagonal squeeze). The details of these correlations are established. Results show that an earlier obtained relationship between \(E\)(phenylene) and \(E\)(hexagonal squeeze) needs to be modified.
Boris Furtula, Slavko Radenković, Ivan Gutman
Bicyclic molecular graphs with the greatest energy
Journal of the Serbian Chemical Society 73 (2008) 431-433
Abstract
The molecular graph \(Q_n\) is obtained by attaching hexagons to the end vertices of the path graph \(P_{n-12}\). Earlier empirical studies indicated that \(Q_n\) has greatest energy among all bicyclic \(n\)-vertex (molecular) graphs. Recently, Li and Zhang proved that \(Q_n\) has greatest energy among all bipartite bicyclic graphs, with the exception of the graphs \(R_{a,b}, a + b = n\), where \(R_{a,b}\) is the graph obtained by joining the cycles \(C_a\) and \(C_b\) by an edge. This result is now completed by showing that \(Q_n\) has the greatest energy among all bipartite bicyclic \(n\)-vertex graphs.
Ivan Gutman, Boris Furtula
Cyclic conjugation in pyracylene
Polycyclic Aromatic Compounds 28 (2008) 136-142
Abstract
Cyclic conjugation of the \(\pi\)-electrons in pyracylene is studied by means of the energy-effects (\(ef\)) of its various cycles. The calculated \(ef\)-values imply that cyclic conjugation significantly destabilizes the pyracylene molecule. This seems to contradict the experimental findings that pyracylene is a reasonably stable conjugated species. We show how this apparent failure of the theory can be avoided, and how the main features of pyracylene's molecular geometry can be rationalized.
Ivan Gutman, Jelena Đurđević, Boris Furtula, Bojana Milivojević
Cyclic conjugation in mono- and dicyclopenta-derivatives of anthracene and phenanthrene
Indian Journal of Chemistry 47A (2008) 803-807
Abstract
The energy effects of cycles and pairs of cycles in mono- and dicyclopenta-derivatives of anthracene and phenanthrene are computed by a graph-theoretical method. Our results show that there is no significant difference between the extent of cyclic conjugation in anthracene and its cyclopenta-derivative (aceanthrylene) and phenanthrene and its cyclopenta-derivative (acephenanthrylene). In contrast to this, the presence of two cyclopentane rings causes drastic changes in the modes of cyclic conjugation. Our results provide an explanation for the peculiar stability order: dicyclopenta[de,mn] anthracene (never isolated) < dicyclopenta[de,kl]anthracene (stable even at \(1000^\circ C\)).
Ivan Gutman, Boris Furtula, Hongbo Hua
Bipartite unicyclic graphs with maximal, second-maximal, and third-maximal energy
MATCH Communications in Mathematical and in Computer Chemistry 58 (2007) 75-82
Abstract
Based on the results of the preceding paper [H. Hua, MATCH Commun. Math. Comput. Chem. 58 (2007) 57-73], by means of an appropriate computer search, the bipartite unicyclic \(n\)-vertex graphs with greatest, second-greatest, and third-greatest energy are determined for all values of \(n\).
Alexandru T. Balaban, Boris Furtula, Ivan Gutman, Radmila Kovačević
Partitioning of \(\pi\)-electrons in rings of aza-derivatives of polycyclic benzenoid hydrocarbons
Polycyclic Aromatic Compounds 27 (2007) 51-63
Abstract
A method is proposed for assessing the \(\pi\)-electron content (\(EC\)) of rings of heteroatom-containing polycyclic conjugated molecules, applicable to the aza-derivatives of benzenoid hydrocarbons. Particular attention is paid to computing \(EC\) values of monoaza-derivatives of acenes. The changes of the \(EC\)-values, relative to those of the parent hydrocarbon, can also be (qualitatively) rationalized by means of resonance-theoretical arguments. General rules are formulated for the effect of substituting one CH group by a nitrogen heteroatom in various positions of an acene.
Ivan Gutman, Sonja Stanković, Jelena Đurđević, Boris Furtula
On the cycle-dependence of topological resonance energy
Journal of Chemical Information and Modeling 47 (2007) 776-781
Abstract
The topological resonance energy (\(TRE\)) was conceived in the 1970s. From the very beginning, it was known that \(TRE\) is equal to the collective energy-effect of all cycles present in a conjugated molecule. Also in the 1970s a theory of cyclic conjugation was elaborated, by means of which it was possible to compute the energy-effect \(ef(Z)\) of each individual cycle \(Z\) present in a conjugated molecule. Yet, the connection between \(TRE\) and the \(ef(Z)\)-values was, until now, not studied. We now show that \(TRE\) and the sum of the \(ef(Z)\)-values are closely correlated but that a certain correction needs to be made by taking into account the energy-effects of pairs, triplets, quartets, etc. of cycles.
Ivan Gutman, Boris Furtula, Biljana Glišić, Violeta Marković, Aleksander Vesel
Estrada index of acyclic molecules
Indian Journal of Chemistry 46A (2007) 723-728
Abstract
A structure-descriptor \(EE\), recently proposed by Estrada, is examined. If \(\lambda_1, \lambda_2,\ldots,\lambda_n\) are the eigenvalues of the molecular graph, then \(EE = \sum_{i=1}^n e^{\lambda_i}\). In the case of trees with \(n\) vertices (that are the graph representations of alkane isomers \(C_nH_{2n+2}\)), the main structural factor influencing the differences between the \(EE\)-values has been found to be the Zagreb index \(Zg\). The coefficient \(b\) in the regression line \(EE = a\,Zg + b\) is an almost perfectly linear function of \(n\), implying that in the case of alkanes, \(EE\) linearly increases with the number of carbon atoms.
Ivan Gutman, Boris Furtula, Radmila Kovačević
Partitioning of \(\pi\)-electrons in rings of aza-derivatives of naphthalene
Journal of the Serbian Chemical Society 72 (2007) 663-671
Abstract
A recently proposed method for calculating the \(\pi\)-electron contents (\(EC\)) of rings of heteroatom-containing polycyclic conjugated molecules was applied to the aza-derivatives of naphthalene. The main finding was that a nitrogen atom in position \(\alpha\) (resp. \(\beta\)) diminishes (resp. increases) the \(EC\)-value of the respective ring. Such a regularity in the displacement of \(\pi\)-electrons can be (qualitatively) rationalized by means of resonance-theoretical reasoning.
Ivan Gutman, Boris Furtula, Violeta Marković, Biljana Glišić
Alkanes with greatest Estrada index
Zeitschrift für Naturforschung 62a (2007) 495-498
Abstract
If \(\lambda_1 \gt \lambda_2 \geq \lambda_3 \geq\ldots\geq\lambda_n\), are the eigenvalues of the molecular graph, then the Estrada index, a recently conceived molecular structure-descriptor is \(EE = \sum_{i=1}^n e^{\lambda_i}\). The same alkanes, whose molecular graphs have extremal Wiener indices and \(\lambda_1\), are shown to be also extremal with regard to the Estrada index.
Ivan Gutman, Slavko Radenković, Boris Furtula, Toufik Mansour, Matthias Schork
Relating Estrada index with spectral radius
Journal of the Serbian Chemical Society 72 (2007) 1321-1327
Abstract
The Estrada index \(EE\) is a recently proposed molecular structure-descriptor, used in the modeling of certain features of the 3D structure of organic molecules, in particular of the degree of folding of proteins and other long-chain biopolymers. The Estrada index is computed from the spectrum of the molecular graph. Therefore, finding its relation with the spectral radius \(r\) (= the greatest graph eigenvalue) is of interest, especially because the structure-dependency of \(r\) is relatively well understood. In this work, the basic characteristics of the relation between \(EE\) and \(r\), which turned out to be much more complicated than initially anticipated, was determined.
Ivan Gutman, Sabina Gojak, Niko Radulović, Boris Furtula
Benzenoid molecules with uniform distribution of \(\pi\)-electrons within rings
Monatshefte für Chemie 137 (2006) 277-284
Abstract
In a recent work it was demonstrated that in linear hexagonal chains the distribution of \(\pi\)-electrons into rings (as computed by means of the Randić–Balaban method) is uniform, irrespective of the nature of the terminal fragments. We now establish that an analogous, yet somewhat more complex, uniformity in the \(\pi\)-electron distribution exists also in double linear hexagonal chains, as well as in some other benzenoid systems.
Ivan Gutman, Andrej Vodopivec, Slavko Radenković, Boris Furtula
On \(\pi\)-electron excess of rings of benzenoid hydrocarbons
Indian Journal of Chemistry 45A (2006) 347-351
Abstract
In a previous paper [Gutman I, Indian J Chem 43A (2004) 1615], the concept of \(\pi\)-electron excess of rings of benzenoid hydrocarbons has been introduced, aimed at amending the \(\pi\)-electron contents of rings computed on the basis of Pauling bond orders. We now show how a \(\pi\)-electron-content-like quantity can be computed from the Hosoya bond orders and establish its close relation with the \(\pi\)-electron excess.
Ivan Gutman, Boris Furtula, Alexandru T. Balaban
Algorithm for simultaneous calculation of Kekulé and Clar structure counts, and Clar number of benzenoid molecules
Polycyclic Aromatic Compounds 26 (2006) 17-35
Abstract
The Chinese mathematicians Heping Zhang and Fuji Zhang conceived a combinatorial polynomial associated with benzenoid molecules. This "Zhang–Zhang polynomial" contains information on the Kekulé structure count (\(K\)), Clar structure count (\(C\)), Clar number (\(Cl\)), Hosoya-Yamaguchi sextet polynomial \(\sigma(B, x)\), and several other important characteristics of the underlying benzenoid molecule. We show how one can easily calculate the Zhang-Zhang polynomial, and thus arrive at an algorithm for simultaneous computation of \(K, C, Cl,\) and \(\sigma(B, x)\). Some applications of this algorithm are pointed out.
Jelena Đurđević, Boris Furtula, Ivan Gutman, Slavko Radenković
Dependence of Hess-Schaad resonance energy on Kekulé structures
Kragujevac Journal of Science 28 (2006) 57-64
Abstract
The Hess-Schaad resonance energy is defined as \(RE = E - E_{ref}\) where \(E\) is the total \(\pi\)-electron energy and \(E_{ref}\) is the reference energy computed by adding double- and single-bond contributions pertaining to some Kekulé structure. Therefore \(RE\) depends on the Kekulé structure considered. We show that in the case of benzenoid molecules the number of only one double-bond type determines \(RE\). Furthermore, \(RE\) is a monotonically increasing function of the number of double bonds of type 3-3 or 2-2, and a monotonically decreasing function of the number of double bonds of type 2-3. This implies that \(RE\) is maximal if the respective Kekulé structure has the greatest possible number of double bonds of type 3-3, or the greatest possible number of double bonds of type 2-2, or the smallest possible number of double bonds of type 2-3.
Ivan Gutman, Sabina Gojak, Boris Furtula, Slavko Radenković, Andrej Vodopivec
Relating total \(\pi\)-electron energy and resonance energy of benzenoid molecules with Kekulé- and Clar-structure-based parameters
Monatshefte für Chemie 137 (2006) 1127-1138
Abstract
Within classes of isomeric benzenoid hydrocarbons various Kekulé- and Clar-structure-based parameters (Kekulé structure count, Clar cover count, Herndon number, Zhang–Zhang polynomial) are all mutually correlated. This explains why both the total \(\pi\)-electron energy (\(E\)), the Dewar resonance energy (\(DRE\)), and the topological resonance energy (\(TRE\)) are well correlated with all these parameters. Nevertheless, there exists an optimal value of the variable of the Zhang–Zhang polynomial for which it yields the best results. This optimal value is negative-valued for \(E\), around zero for \(TRE\), and positive-valued for \(DRE\). A somewhat surprising result is that \(TRE\) and \(DRE\) considerably differ in their dependence on Kekulé- and Clar-structure-based parameters.
Ivan Gutman, Boris Furtula
Equivalence of two models for partitioning of \(\pi\)-electrons in rings of benzenoid hydrocarbons
Zeitschrift für Naturforschung 61a (2006) 281-285
Abstract
Two recently proposed methods for assessing the \(\pi\)-electron contents of rings of benzenoid hydrocarbons are shown to be equivalent.
Ivan Gutman, Nedžad Turković, Boris Furtula
On distribution of \(\pi\)-electrons in rhombus-shaped benzenoid hydrocarbons
Indian Journal of Chemistry 45A (2006) 1601-1604
Abstract
The distribution of \(\pi\)-electrons into rings of rhombus-shaped benzenoid hydrocarbons is reported here. In this class of benzenoid systems, the (unique) Clar formula represents only a minute fraction of the total number of Kekule structures, and therefore a breakdown of the Clar model could be expected. It is found that the \(\pi\)-electron distribution follows a pattern different from what is predicted by the Clar model : the greatest \(\pi\)-electron content is in the two "full" peak hexagons (around 4.5 electrons). In the other boundary hexagons, the \(\pi\)-electrons are distributed almost uniformly (around 3 electrons per hexagon). In the internal hexagons, the \(\pi\)-electrons are also distributed in an almost uniform manner (around 2 electrons per hexagon), the \(\pi\)-electron content of the "full" hexagons insignificantly exceeding the \(\pi\)-electron contents of the "empty" hexagons.
Ivan Gutman, Boris Furtula
A Kekulé structure basis for phenylenes
Journal of Molecular Structure (Theochem) 770 (2006) 67-71
Abstract
By means of Kekulé structures and, in particular, their count \(K\), many properties of benzenoid molecules can be rationalized. The analogous properties of non-benzenoid polycyclic conjugated molecules require the consideration of some, but not all, Kekulé structures, whose number is the precisely defined and long time known ‘algebraic structure count’ \(ASC\). In the general case it is not known how to construct a Kekulé structure basis for non-benzenoid molecules, consisting of \(ASC\) Kekulé structures, \(ASC \leq K\). We now offer a solution of this problem for phenylenes and point out some applications.
Boris Furtula, Ivan Gutman
Assessing the distribution of \(\pi\)-electrons into rings of phenylenes
Indian Journal of Chemistry 45A (2006) 1977-1980
Abstract
We propose a novel method for assessing the distribution of \(\pi\)-electrons into rings of phenylenes. This method is equivalent to, but significantly simpler than, the original approach of Randić and Balaban.
Ivan Gutman, Boris Furtula, Nedžad Turković
Electron and energy contents of hexagons in benzenoid hydrocarbons
Polycyclic Aromatic Compounds 25 (2005) 87-94
Abstract
In analogy to the recently introduced concept of \(\pi\)-electron content of a ring (\(EC\)), that is calculated from the Pauling bond orders, we define the \(\pi\)-electron energy content of the ring (\(ec\)), calculated in an analogous manner from the Coulson bond orders. The relations between \(EC\) and \(ec\) are analyzed for the rings of catacondensed benzenoid hydrocarbons.
Ivan Gutman, Boris Furtula, Alexander A. Toropov, Alla P. Toropova
The graph of atomic orbitals and its basic properties. 2. Zagreb indices
MATCH Communications in Mathematical and in Computer Chemistry 53 (2005) 225-230
Abstract
Relations are established between the first and second Zagreb index (\(M_{1}\), and \(M_{2}\)) of the graph of atomic orbitals (\(G^{*}\)), and of the ordinary hydrogen-depleted (\(G(C)\)) and hydrogen-filled (\(G(H)\)) molecular graphs of saturated hydrocarbons. It is shown that \(M_1(G^{*}) = 12 M_1(G(C)) + 9 M_1(G(H)) + 96(c - 1)\) and \(M_2(G^{*}) = 36M_2(G(C)) + 15M_2(G(H)) + 54M_1(G(C)) + 96(c - 1)\), where \(c\) is the number of (independent) cycles of the underlying molecule.
Alexander A. Toropov, Ivan Gutman, Boris Furtula
Graph of atomic orbitals and molecular structure-descriptors based on it
Journal of the Serbian Chemical Society 70 (2005) 669-674
Abstract
The graph of atomic orbitals (GAO) is a novel type of molecular graph recently proposed by one of the authors. Various molecular structure-descriptors computed for GAO are compared with their analogs computed for ordinary molecular graphs. The quality of these structure-descriptors was tested for correlation with the normal boiling points of alkanes and cycloalkanes. In all the studied cases, the results based on GAO are similar to, and usually slightly better than, those obtained by means of ordinary molecular graps.
Ivan Gutman, Sabina Gojak, Nedžad Turković, Boris Furtula
Polansky's benzenoid character and the electron content of rings of benzenoid hydrocarbons
MATCH Communications in Mathematical and in Computer Chemistry 53 (2005) 139-145
Abstract
In 2004 Balaban and Randić put forward a method to assess the \(\pi\)-electron content \(EC\) of rings of benzenoid hydrocarbons. Much earlier, in the 1960s, Oskar Polansky conceived and elaborated the concept of benzenoid character \(\rho\) of the same rings. We show that \(EC\) and \(\rho\) are linearly correlated, implying that the \(\pi\)-electron content of a ring is just another way of expressing its benzenoid character (or vice versa). The fine details of the correlation between \(EC\) and \(\rho\) are established.
Boris Furtula, Ivan Gutman, Nedžad Turković
Relation between electron and energy contents of hexagons in catacondensed benzenoid hydrocarbons
Indian Journal of Chemistry 44A (2005) 9-12
Abstract
The concept of electron content (\(EC\)) of hexagons in benzenoid hydrocarbons has been recently introduced in a series of scientific papers. In full analogy to it one may conceive also the energy content (\(ec\)) of hexagons. These contents are mutually related, but not in a manner that could be anticipated. On the basis of the \(EC\)- and \(ec\)-values calculated for a large number of catacondensed benzenoid hydrocarbons we establish the actual relation between these quantities. Within hexagons of the same type (terminal, linearly annelated, angularly annelated, and branched) the relation between \(ec\) and \(EC\) is nearly linear and the respective regression lines are nearly parallel and equidistant.
Ivan Gutman, Svetlana Milosavljević, Boris Furtula, Nataša Cmiljanović
Relation between electron and energy contents of hexagons in pericondensed benzenoid hydrocarbons
Indian Journal of Chemistry 44A (2005) 13-17
Abstract
Relations are established between the \(\pi\)-electron content (\(EC\)) and the \(\pi\)-electron energy electron energy content (\(ec\)) of hexagons in pericondensed benzenoid hydrocarbons. Whereas in catacondensed benzenoids only four types of differen tly annelated hexagons need to be distinguished [Furtula, Gutman & Turković, Indian J Chem 43A (2004)], in pericondensed systems there are 12 different annelation types. We show that within each of the 12 classes of equally annelated hexagons there exists a linear correlation between \(EC\) and \(ec\). These correlations can be explained by means of the Cruickshank-Sparks equations, which relate the Pauling and Coulson bond orders. In addition, within each annelation class there is a correlation between the \(ec\)-value of a hexagon and the effect \(ef\) of the same hexagon on the total \(\pi\)-electron energy of the corresponding benzenoid molecule. Most of these latter correlations are curvilinear. Our main conclusions are that (a) \(ec\) is proportional to \(EC\) and (b) \(ec\) is proportional to \(ef\), but (c) the mode of the annelation of the respective hexagon strongly influences the actual form of these interdependencies.
Ivan Gutman, Boris Furtula, Veselin Vučković, Biljana Arsić, Marijan Ranđelović
Partition of \(\pi\)-electrons in rings of double linear hexagonal chains
Bulletin de l'Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 130 (2005) 97-105
Abstract
Two methods for assessing the \(\pi\)-electron content of a ring of a benzenoid hydrocarbon were recently proposed: one based on the averaging of the \(\pi\)-electron contents of the Kekulé structural formulas, and the other based on an analogous treatment of the Clar aromatic sextet formulas. We apply these two methods to the homologous series consisting of two condensed linear polyacene units (whose first members are pyrene, anthanthrene, peri-naphthacenonaphthacene, ...). The two approaches give essentially the same results. Contrary to the case of linear polyacenes (in which the partition of \(\pi\)-electrons into rings is uniform), in their double-chain analogs the partition of \(\pi\)-electrons is found to be highly non-uniform. The electron contents monotonically decrease along each polyacene chain, being maximal in the hexagons having the smallest number (= 2) of neighbors. Several other regularities of the \(\pi\)-electron distribution are also established.
Ivan Gutman, Lemi Türker, Boris Furtula, Veselin Vučković
The McClelland number of conjugated hydrocarbons
Croatica Chemica Acta 78 (2005) 485-488
Abstract
The McClelland number of a conjugated hydrocarbon is the integer \(k\), satisfying the condition \(2^{-(1/2)^k} \sqrt{2nm} \leq E \lt 2^{–(1/2)^{k+1}}\sqrt{2nm}\), where \(E\) is the HMO total \(\pi\)-electron energy, \(n\) the number of carbon atoms, and \(m\) the number of carbon-carbon bonds. If \(k = 3\), then the respective conjugated system is said to be energy-regular. If \(k \leq 2\) and \(k \geq 4\), then one speaks of energy-poor and energy-rich \(\pi\)-electron systems, respectively. We found that all polycyclic Kekuléan hydrocarbons, possessing condensed rings, are energy-regular, with only three exceptions: naphthalene, phenanthrene, and triphenylene (which are energy-rich). Energy-poor \(\pi\)-electron systems are some (but not all) non-Kekuléans, whereas many of the polycyclic Kekuléan hydrocarbons with non-condensed rings (polyphenyls, phenyl-substituted polyenes and similar) are energy-rich.
Ivan Gutman, Slavko Radenković, Boris Furtula, Haruo Hosoya
Some properties of the topological bond order
Chemical Physics Letters 407 (2005) 73-77
Abstract
The topological bond order is a bond-order-like quantity, put forward in the 1970s. It is defined as \(p_{rs}^T = Z(G_{rs})/Z(G)\), where \(G\) is the molecular graph, \(G_{rs}\) is obtained from \(G\) by deleting from it the adjacent vertices labelled by \(r\) and \(s\), and \(Z\) stands for the respective topological (Hosoya) index. Because no easy way for the calculation of \(p_{rs}^T\) is known, its properties were studied only to a limited degree. We now introduce a modified topological bond order, \(\tilde{p}_{rs}^T\), that can (easily) be calculated from the eigenvalues of \(G\) and \(G_{rs}\). For acyclic systems, \(\tilde{p}_{rs}^T = p_{rs}^T\). In the case of polycyclic systems a reasonably accurate linear correlation exists between \(\tilde{p}_{rs}^T\) and \(p_{rs}^T\). Thus, by studying \(\tilde{p}_{rs}^T\) the main properties of \(p_{rs}^T\) can be stablished.
Ivan Gutman, Boris Furtula, Jelena Đurđević, Radmila Kovačević, Sonja Stanković
Annelated perylenes: Benzenoid molecules violating the Kekulé-structure-based cyclic conjugation models
Journal of the Serbian Chemical Society 70 (2005) 1023-1031
Abstract
Several currently used models for assessing the extent of cyclic conjugation in benzenoid hydrocarbons, all based on Kekulé-type structural formulas, predict that there is no cyclic conjugation in the central, "empty", ring of perylene and its annelated derivatives. In this paper it is shown that in some annelated perylenesthe cyclic conjugation in the "empty" ring (measured by its energy-effect) may be unexpectedly high. Therefore, in the case of these annelated perylenes, the Kekulé-structure-based models fail. The cause for such an “anomalous” behavior of annelated perylenes is discussed.
Ivan Gutman, Milan Randić, Alexandru T. Balaban, Boris Furtula, Veselin Vučković
\(\pi\)-Electron contents of rings in the double-hexagonal-chain homologous series (pyrene, anthanthrene, and other acenoacenes)
Polycyclic Aromatic Compounds 25 (2005) 215-226
Abstract
Recently three methods for calculating the \(\pi\)-electron content of rings of benzenoid hydrocarbons were put forward: one based on the consideration of Kekulé structural formulas, and the other two based on an analogous treatment of the Clar aromatic sextet formulas. These three methods are applied to the homologous series consisting of two condensed acene chains (whose first members are pyrene, anthanthrene, peri-naphthacenonaphthacene, ...), leading to basically identical results. In contrast to acenes (in which the partition of \(\pi\)-electrons into rings is uniform), in the double-hexagonal-chain species the partition of \(\pi\)-electrons is highly non-uniform. The electron content monotonically decreases, in opposite directions, along the two acene chains, being maximal in the least annelated rings. Some other generally valid regularities in the \(\pi\)-electron properties of the double–hexagonal–chain benzenoids are also pointed out.
Ivan Gutman, Boris Furtula, Svetlana Jeremić, Nedžad Turković
Electron content of rings of fully benzenoid hydrocarbons
Journal of the Serbian Chemical Society 70 (2005) 1199-1204
Abstract
The distribution of \(\pi\)-electrons in rings of fully benzenoid hydrocarbons was investigated. It was found that the electron content \(EC\) of "full" rings varies significantly (between 5.5 and 2.5 electrons), and depends on the annelation mode, mainly on the number of adjacent rings. "Full" rings belonging to the same annelation class have nearly equal \(EC\)-values. The \(EC\)-values of all "empty" rings are also nearly equal (around 2 electrons).
Ivan Gutman, Sonja Stanković, Radmila Kovačević, Jelena Đurđević, Boris Furtula
Anomalous cyclic conjugation in benzenoid molecules with a small number of Kekulé structures
Indian Journal of Chemistry 44A (2005) 1751-1754
Abstract
The currently used Kekulé-structure-based models for assessing the extent of cyclic conjugation in benzenoid hydrocarbons predict that there is no or very little cyclic conjugation in the rings possessing "fixed" double and single carbon-carbon bonds. With the example of a suitably chosen class of benzenoid systems with a small number of Kekulé structures, we show that the cyclic conjugation in the rings with "fixed" bonds (measured by its energy-effect) may be unexpectedly high, contradicting the results of standard Kekulé-structure-based considerations. Consequently, our study reveals that the analysis of the conjugation modes of polycyclic aromatic compounds, based solely on Kekulé structures, may sometimes be insufficient and may lead to erroneous conclusions.
Ivan Gutman, Sabina Gojak, Boris Furtula
Clar theory and resonance energy
Chemical Physics Letters 413 (2005) 396-399
Abstract
A mathematical model, referred here as the Zhang–Zhang polynomial \(\zeta(x)\), that embraces all the main concepts encountered in the Clar aromatic sextet theory of benzenoid hydrocarbons, was recently put forward by Zhang and Zhang. We now show that \(\zeta(x)\) is related to resonance energy, and that \(\ln{\zeta(x)}\) and \(RE\) are best correlated when $x \approx 1$. This indicates that \(\zeta(1)\) could be viewed as a (novel) structure-descriptor, playing a role analogous to the Kekulé structure count in Kekulé-structure-based theories. Some basic properties of \(\zeta(1)\) are established.
Ivan Gutman, Sabina Gojak, Sonja Stanković, Boris Furtula
A concealed difference between the structure-dependence of Dewar and topological resonance energy
Journal of Molecular Structure (Theochem) 757 (2005) 119-123
Abstract
In the 1970s, two Dewar-type resonance energies were put forward, one by Hess and Schaad (denoted here by \(DRE\)), the other by Gutman et al. and, independently, by Aihara (denoted here by \(TRE\)). Both were believed to be equivalent and, indeed, a good linear correlation between them could be established. We now show that there are some significant differences between the structure-dependencies of \(DRE\) and \(TRE\). In particular, in the case of benzenoid molecules, \(DRE\) and \(TRE\) are found to be linearly related to \(\ln{\zeta(0)}\) and \(\ln{\zeta(1)}\), respectively, where \(\zeta(x)\) is the Zhang-Zhang polynomial. Whereas \(\zeta(0)\) is just the Kekulé structure count, \(\zeta(1)\) is a novel structure-descriptor overlooked in the previous studies of resonance energy.
Ivan Gutman, Boris Furtula, Dušica Vidović, Haruo Hosoya
A concealed property of the topological index Z
Bulletin of the Chemical Society of Japan 77 (2004) 491-496
Abstract
Examination of the structure-dependence of the total \(\pi\)-electron energy leads to the equation \(F(G,x) = \ln{Z(G)}\), where \(F(G,x)\) is the Coulson function (of the molecular graph \(G\)) and \(Z(G)\) is the corresponding topological index \(Z\). The (positive and unique) solution of this equation is called the \(Z\)-point of \(G\) and is denoted by \(x_H\). By the analysis of the \(Z\)-points of trees and chemical trees the following generally valid regularities were established: (a) Not all trees have a \(Z\)-point, but all chemical trees have a \(Z\)-point. (b) The \(Z\)-points of all chemical trees (irrespective of their size and other structural features) are nearly equal; for all chemical trees, \(x_H \approx 1.2\).
Ivan Gutman, Boris Furtula, Biljana Arsić, Žarko Bošković
On the relation between Zenkevich and Wiener indices of alkanes
Journal of the Serbian Chemical Society 69 (2004) 265-271
Abstract
A relatively complicated relation was found to exist between the quantity \(U\) recently introduced by Zenkevich (providing a measure of internal molecular energy), and the Wiener index \(W\) (measuring molecular surface area and intermolecular forces). We now report a detailed analysis of this relation and show that, in the case of alkanes, its main features are reproduced by the formula \(U = –\alpha W + \beta + \gamma \nu_1\); where \(\nu_1\) is the number of methyl groups, and \(\alpha\), \(\beta\) and \(\gamma\) are constants, depending only on the number of carbon atoms. Thus for isomeric alkanes with the same number of methyl groups, \(U\) and \(W\) are linearly correlated.
Ivan Gutman, Boris Furtula, Damir Vukičević, Biljana Arsić
Equiseparable molecules and molecular graphs
Indian Journal of Chemistry 43A (2004) 7-10
Abstract
By deleting an edge \(e_i\) from the molecular graph \(G\) of an acyclic molecule, it decomposes into two fragments, with \(n_1(e_i|G)\) and vertices, \(n_1(e_i|G) \leq n_2(e_i|G)\) \(i = 1, 2,\ldots,m\). If the edges of the graphs \(G'\) and \(G''\), representing two isomeric acyclic molecules, can be chosen so that \(n_1(e_i│G') = n_1(e_i│G'')\) holds for all \(i = 1,2, \ldots,m\), then \(G'\) and \(G''\) are said to be equiseparable. The respective two isomeric molecules are also said to be equiseparable. The main (known) physicochemical consequences of equiseparability are pointed out. A general method for designing pairs of equiseparable molecular graphs is described, by which large sets of equiseparable species can be constructed.
Ivan Gutman, Boris Furtula, Olga Miljković, Marija Rakić
Families of equiseparable trees and chemical trees
Kragujevac Journal of Science 26 (2004) 19-30
Abstract
Let \(T\) be an \(n\)-vertex tree and \(e\) its edge. By \(n_1(e|T)\) and \(n_2(e|T)\) are denoted the number of vertices of \(T\) lying on the two sides of \(e\); \(n_1(e|T) + n_2(e|T) = n\). Conventionally, \(n_1(e|T) \leq n_2(e|T)\). If \(T'\) and \(T''\) are two trees with the same number \(n\) of vertices, and if their edges \(e'_1, e'_2,\ldots, e'_{n-1}\) and \(e''_1, e''_2,\ldots,e''_{n-1}\) can be labeled so that \(n_1(e'_i|T') = n_1(e''_i|T'')\) holds for all \(i = 1, 2,\ldots, n- 1\), then \(T'\) and \(T''\) are said to be equi-separable. There exist large families of equiseparable trees. We report here the results of a systematic study of these families for \(7 \leq n \leq 20\).
Olga Miljković, Boris Furtula, Ivan Gutman
Statistics of equiseparable trees and chemical trees
MATCH Communications in Mathematical and in Computer Chemistry 51 (2004) 179-184
Abstract
If \(T\) is a tree and \(e\) its edge, then $T - e$ consists of two components, with \(n_1(e|T)\) and \(n_2(e|T)\) vertices. Conventionally, \(n_1(e|T) \leq n_2(e|T)\). If \(T'\) and \(T''\) are two trees of the same order \(n\), and if their edges \(e′_1, e′_2,\ldots,e′_{n-1}\) and \(e''_1, e''_2,\ldots,e''_{n-1}\) can be labeled so that \(n_1(e′_i|T') = n_1(e''_i|T'')\) holds for all \(i = 1,2,\ldots,n-1\), then \(T'\) and \(T''\) are said to be equiseparable. We examined \(n\)-vertex trees and chemical trees (\(7 \leq n \leq 20\)) with regard to equiseparability. There exist very large families of equiseparable trees and chemical trees, and only a relatively few of them have no equiseparable mate.
Ivan Gutman, Boris Furtula
Effect of ethyl groups on the relation between Zenkevich and Wiener indices
Kragujevac Journal of Science 26 (2004) 31-42
Abstract
The Zenkevich index \(U\) provides a measure of the intramolecular energy of an organic molecule. The Wiener index \(W\) represents the surface area of the same molecules. Thus, from a physico-chemical point of view, no connection between \(U\) and \(W\) would be anticipated. Yet, in the case of alkanes. these two quantities depend on the same structural features of the molecular graph and an approximate mathematical relation between them can be established. Within sets of isomeric alkanes, the relation between \(U\) and \(W\) is linear, the \((U W)\)-points forming several, mutually parallel, lines. Each such line pertains to a group of isomers possessing a fixed number of methyl and ethyl groups.
Ivan Gutman, Boris Furtula, Biljana Arsić
On structure descriptors related with intramolecular energy of alkanes
Zeitschrift für Naturforschung 59a (2004) 694-698
Abstract
In an earlier work it was demonstrated that the Zenkevich index \(U\) provides a measure of the intramolecular energy of an organic molecule, and that - in the case of alkanes - it is related to the Wiener index. We now show that \(U\) is much closer related to the recently introduced variable Wiener index \(W_\lambda\): Within sets of isomeric alkanes, the relation between \(U\) and \(W\); is linear, the \((U, W_\lambda)\)-points forming several, mutually parallel, lines. Each such line pertains to a group of isomers possessing a fixed number of methyl groups. There exists a critical value of the parameter; for which all the \((U, W_\lambda)\)-lines coalesce, in which case the relation between \(U\) and \(W_\lambda\) becomes independent of the number of methyl groups. Approximate analytical expressions for the \((U, W_\lambda)\)-dependence are deduced.
Ivan Gutman, Boris Furtula, Slavko Radenković
Relation between Pauling and Coulson bond orders in benzenoid hydrocarbons
Zeitschrift für Naturforschung 59a (2004) 699-704
Abstract
The relation between Pauling and Coulson bond orders in benzenoid hydrocarbons is examined. The carbon'carbon bonds of benzenoid hydrocarbons have to be classified into three classes, depending on the number of attached hydrogen atoms. Within each class the correlation between the bond orders is linear. The results can be used to rationalize the recently discovered correlation between the energy and electron contents of rings. An approximate expression for the total \(\pi\)-electron energy is also deduced.
Ivan Gutman, Dušica Vidović, Boris Furtula
Chemical applications of the Laplacian spectrum. VII. Studies of the Wiener and Kirchhoff indices
Indian Journal of Chemistry 42A (2003) 1272-1278
Abstract
Some further chemical applications of the Laplacian spectra are reported. The Kel'mans theorem for the calculation of the coefficients of the Laplacian characteristic polynomial is stated and exemplified. By means of this theorem a (previously known) formula for the Wiener and Kirchhoff index is deduced. It is shown that the Wiener index is correlated with the "algebraic connectivity", namely, the smallest positive Laplacian eigenvalue. Lower and upper bounds for the Kirchhoff index are obtained.
Ivan Gutman, Dušica Vidović, Boris Furtula, Igor G. Zenkevich
Wiener-type indices and internal molecular energy
Journal of the Serbian Chemical Society 68 (2003) 401-408
Abstract
In earlier studies it was established that internal molecular energies (\(E_{int}\)) of alkanes can be reproduced, in an approximate yet reliable manner by means of a molecular-graph-based structure-descriptor \(U\). It was also established that \(U\) is linearly correlated with the Wiener index \(W\). We now show that the correlation between \(U\) and \(W\) is more complicated than earlier expected, and that it cannot be represented by a single line. We also show that a very good linear correlation exists between U and a modified version \(W_m(\lambda)\) of the Wiener index, which is thus more suitable for modeling Eint than the ordinary Wiener index.
Ivan Gutman, Dušica Vidović, Boris Furtula, Aleksander Vesel
Relations between topological indices of large chemical trees
Indian Journal of Chemistry 42A (2003) 1241-1245
Abstract
Several approximate relations have recently been established between molecular-graph-based structure descriptors of alkanes, in particular between (a) eigenvalue sum and Hosoya index. (b) greatest graph eigenvalue and connectivity index, (c) Wiener index and smallest positive Laplacian eigenvalue, (d) greatest Laplacian and greatest ordinary graph eigenvalue, (e) Zenkevich and Wiener index, and (f) hyper-Wiener and Wiener index. These all have been found to hold for alkanes with \(n = 10\) or fewer carbon atoms, and have verified on samples consisting of all alkane isomers. Applying an algorithm for generating trees uniformly by random we have now tested these regularities for very large chemical trees (\(n = 50\)). It has been found that regularities (c) and (f) hold equally well in the case of very large chemical trees, whereas regularities (a), (d) and (e) are applicable, but with significantly attenuated accuracy. Regularity (b) vanishes at large values of n.
Ivan Gutman, Boris Furtula
Hyper-Wiener index vs. Wiener index. Two highly correlated structure-descriptors
Monatshefte für Chemie 134 (2003) 975-981
Abstract
The Wiener (\(W\)) and hyper-Wiener (\(WW\)) indices of alkanes are found to be highly correlated. Hence, these two structure-descriptors pertain to the very same structural features of the underlying molecules and one of them may be viewed as superfluous. For alkane isomers with \(n\) carbon atoms, \(WW\) is bounded from both above and below by linear functions of \(W\). The upper bound \((n/4+2)W − n(n−1)(n+1)/4\) and the lower bound \((3/2)W − (n−1)/2\) for \(W \leq W_0\) and \((3n/4)W − n(n−1)^2(n+1)\) for \(W \geq W_0\), where \(W_0=(2/3)(n−1)(n^3−n−1)/(n−2)\) are better than the previously reported estimates of the same kind. In spite of this, the correlation between \(W\) and \(WW\) is curvilinear.
Ivan Gutman, Biljana Arsić, Boris Furtula
Equiseparable chemical trees
Journal of the Serbian Chemical Society 68 (2003) 549-555
Abstract
Let \(n_1(e|T)\) and \(n_2(e|T)\) denote the number of vertices of a tree \(T\), lying on the two sides of the edge \(e\). Let \(T_1\) and \(T_2\) be two trees with equal number of vertices, let \(e\) be an edge of \(T_1\) and \(f\) an edge of \(T_2\). Then \(e\) and \(f\) are said to be equiseparable if either \(n_1(e|T_1) = n_1(f|T_2)\) or \(n_1(e|T_1) = n_2(f|T_2)\). If all edges of \(T_1\) and \(T_2\) can be chosen so as to form equiseparable pairs, then \(T_1\) and \(T_2\) are equiseparable trees. A number of molecular structure-descriptors of equiseparable chemical trees coincide, implying that the corresponding alkane isomers must have similar physico-chemical properties. It is shown how equiseparable chemical trees can be constructed in a systematic manner.
Ivan Gutman, Boris Furtula, Jasmina Belić
Note on the hyper-Wiener index
Journal of the Serbian Chemical Society 68 (2003) 943-948
Abstract
The hyper-Wiener index \(WW\) of a chemical tree \(T\) is defined as the sum of the products \(n_1n_2\), over all pairs \(u,v\) of vertices of \(T\), where \(n_1\) and \(n_2\) are the number of vertices of \(T\), lying on the two sides of the path which connects \(u\) and \(v\). We examine a slight modification \(WWW\) of the hyper-Wiener index, defined as the sum of the products \(n_1n_2n_3\), over all pairs \(u,v\) of vertices of \(T\), where \(n_3\) is the number of vertices of \(T\), lying between \(u\) and \(v\). It is found that \(WWW\) correlates significantly better with various physico-chemical properties of alkanes than \(WW\). Lower and upper bounds for \(WWW\), and an approximate relation between \(WWW\) and \(WW\) are obtained.
Ivan Gutman, Dušica Vidović, Boris Furtula
Coulson function and Hosoya index
Chemical Physics Letters 355 (2002) 378-382
Abstract
Let \(G\) be a molecular graph, n the number of its vertices and \(\phi(G,x)\) its characteristic polynomial. Already in 1940 Coulson expressed the total \(\pi\)-electron energy of conjugated unsaturated molecules in terms of the function \(F(x)=n - ix\phi'(G,ix)/\phi(G,ix)\). Recently, the Coulson function \(F(x)\) found applications also in modeling the structure-dependence of physico-chemical properties of alkanes. We now analyze the Coulson function and establish some of its hitherto unnoticed features, in particular its relations with the Hosoya index.
Ivan Gutman, Boris Furtula, Dušica Vidović
Coulson function and total \(\pi\)-electron energy
Kragujevac Journal of Science 24 (2002) 71-82
Abstract
It is shown that the value of the variable \(x\), at which the Coulson function \(F(G,x)\) has its inflection point, is related to the respective total \(\pi\)-electron energy (\(E\)) and depends mainly on \(E\) and the number of non-bonding molecular orbitals.
Boris Furtula, Ivan Gutman, Željko Tomović, Aleksander Vesel, Igor Pesek
Wiener-type topological indices of phenylenes
Indian Journal of Chemistry 41A (2002) 1767-1772
Abstract
The Wiener index \(W\) of a phenylene (\(PH\)) has earlier been shown to depend in a mathematically exact manner on the Wiener indices of its hexagonal squeeze (\(HS\)) and inner dual (\(ID\)). We now examine the analogous dependency for a class of Wiener-type indices \(W_\lambda\), where \(\lambda\) is a variable parameter (\(W_\lambda = W\) for \(\lambda = 1\)) is examined now. It is shown that some features of the dependency of \(W(PH)\) on \(W(HS)\) and \(W(ID)\) arc maintained for all values of \(\lambda\), \(-5 \leq \lambda \leq +5\), whereas some are violated.