On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter

The total number of matchings of a graph is defined as its Hosoya index. Conjugated and non-conjugated acyclic graphs that have maximal Hosoya index and short diameter are characterized in this paper, explicit expressions of the Hosoya indices of these extremal graphs are also presented.     

Novel shape descriptors for molecular graphs.

We report on novel graph theoretical indices which are sensitive to the shapes of molecular graphs. In contrast to the Kier's kappa shape indices which were based on a comparison of a molecular graph with graphs representing the extreme shapes, the linear graph and the "star" graph, the new shape indices are obtained by considering for all atoms the number of paths and the number of walks within a graph and then making the quotients of the number of paths and the number of walks the same length. The new shape indices show much higher discrimination among isomers when compared to the kappa shape indices. We report the new shape indices for smaller alkanes and several cyclic structures and illustrate their use in structure-property correlations. The new indices offer regressions of high quality for diverse physicochemical properties of octanes. They also have lead to a novel classification of physicochemical properties of alkanes.     

CMMSE-on the first general Zagreb index

The aim of this paper is to obtain new inequalities involving the first general Zagreb index, and characterize graphs which are extremal with respect to them. We also obtain inequalities involving the forgotten and second general Zagreb indices.     

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Pe?a gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Pe?a are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.     

Mostar index

We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic graphs. Then we show how it can be efficiently computed for large classes of chemically interesting graphs using a variant of the cut method introduced by Klav?ar, Gutman and Mohar. Explicit formulas are presented for several classes of benzenoid graphs and Cartesian products. At the end we state several conjectures and list some open problems.     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

On the difference between atom‐bond connectivity index and Randić index of binary and chemical trees

This article is devoted to establishing some extremal results with respect to the difference of two well‐known bond incident degree indices [atom‐bond connectivity (ABC ) index and Randi? (R ) index] for the chemical graphs representing alkanes. More precisely, the first three extremal trees with respect to ABC – R are characterized among all n‐vertex binary trees (the trees with maximum degree at most 3). The n‐vertex chemical trees (the trees with maximum degree at most 4) having the first three maximum ABC – R values are also determined.     

Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs

Given a collection of connected graphs one may build bridge and chain graphs out of them. In this paper it is shown how the Wiener, hyper-Wiener, detour and hyper-detour indices for bridge and chain graphs are determined from the respective indices of the individual graphs. The results obtained are illustrated by some examples.     

Stereotopological indices for a family of chemical graphs

We describe a mathematical method that can be employed to define stereotopological indices of placements of certain graphs in space. These indices are applied to successfully distinguish between configurations in a chemically interesting family of knotted and/or linked four-valent oriented graphs in space. The methods are fundamentally algebraic and combinatorial in nature and are most readily understood in the context of calculations and the study of several key examples that are presented.     

Complete graph conjecture for inner-core electrons: homogeneous index case

The complete graph conjecture that encodes the inner-core electrons of atoms with principal quantum number n >or= 2 with complete graphs, and especially with odd complete graphs, is discussed. This conjecture is used to derive new values for the molecular connectivity and pseudoconnectivity basis indices of hydrogen-suppressed chemical pseudographs. For atoms with n = 2 the new values derived with this conjecture are coincident with the old ones. The modeling ability of the new homogeneous basis indices, and of the higher-order terms, is tested and compared with previous modeling studies, which are centered on basis indices that are either based on quantum concepts or partially based on this new conjecture for the inner-core electrons. Two similar algorithms have been proposed with this conjecture, and they parallel the two "quantum" algorithms put forward by molecular connectivity for atoms with n > 2. Nine properties of five classes of compounds have been tested: the molecular polarizabilities of a class of organic compounds, the dipole moment, molar refraction, boiling points, ionization energies, and parachor of a series of halomethanes, the lattice enthalpy of metal halides, the rates of hydrogen abstraction of chlorofluorocarbons, and the pED(50) of phenylalkylamines. The two tested algorithms based on the odd complete graph conjecture give rise to a highly interesting model of the nine properties, and three of them can even be modeled by the same set of basis indices. Interesting is the role of some basis indices all along the model.     

QSPR modeling: graph connectivity indices versus line graph connectivity indices

Five QSPR models of alkanes were reinvestigated. Properties considered were molecular surface-dependent properties (boiling points and gas chromatographic retention indices) and molecular volume-dependent properties (molar volumes and molar refractions). The vertex- and edge-connectivity indices were used as structural parameters. In each studied case we computed connectivity indices of alkane trees and alkane line graphs and searched for the optimum exponent. Models based on indices with an optimum exponent and on the standard value of the exponent were compared. Thus, for each property we generated six QSPR models (four for alkane trees and two for the corresponding line graphs). In all studied cases QSPR models based on connectivity indices with optimum exponents have better statistical characteristics than the models based on connectivity indices with the standard value of the exponent. The comparison between models based on vertex- and edge-connectivity indices gave in two cases (molar volumes and molar refractions) better models based on edge-connectivity indices and in three cases (boiling points for octanes and nonanes and gas chromatographic retention indices) better models based on vertex-connectivity indices. Thus, it appears that the edge-connectivity index is more appropriate to be used in the structure-molecular volume properties modeling and the vertex-connectivity index in the structure-molecular surface properties modeling. The use of line graphs did not improve the predictive power of the connectivity indices. Only in one case (boiling points of nonanes) a better model was obtained with the use of line graphs.     

Novel graph-theoretical approach to estimating the relative importance of individual Kekulé valence structures. II. Benzenoid systems having four to seven fused benzene rings: “Pseudodegenerate” states

Kekulé structures of 10 nonlinear acenes comprising 83 graphs are studied through the use of connectivities [M. Randi?, J. Am. Chem. Soc. 97 , 6609 (1975)] of their corresponding submolecules [H. Joela, Theor. Chim. Acta 39 , 241 (1975)]. In certain rare cases states were identified to have identical branching indices but different Kekulé indices [A. Graovac, I. Gutman, M. Randi?, and N. Trinajsti?, J. Am. Chem. Soc. 95 , 6267 (1973)]. Such states are termed pseudodegenerate states. A method is described to forecast and another to remedy such situations. The method emphasizes the relation between VB (resonance) and MO theories using graph-theoretical concepts.     

On the ABC index of connected graphs with given degree sequences

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.     

Establishment of correlations between the structure and reactivity of molecules in the gas phase based on information theory

A general approach to revealing correlations between the structure of molecules and their reactivity in fragmentation processes under electron impact conditions based on the use of generalized structural and mass spectral characteristics is suggested. The characteristics were obtained using information theory, molecular graphs, and absolute reaction rates. Information topological indices of molecular graphs were used as generalized structural characteristics of molecules. They are a quantitative measure of the structural complexity of molecules and are expressed in information units. The gas-phase process of fragmentation of molecules under electron impact was used as a general reaction series for all volatiles. In terms of information theory, the mass spectrum represents the distribution of probabilities of the formation of ions of each type, and the information entropy of this distribution appears to be an integral characteristic of the reactivity of a molecule during fragmentation under electron impact in the gas phase. Using organic and organometallic compounds of several classes (ferrocene derivatives, arylsilanes, aromatic azo compounds,etc.) as examples, linear correlations between the information indices of the mass spectra and the information topological indices of the appropriate molecular graphs or electronic parameters of molecules have been found, which testifies that the approach suggested is adequate.Translated fromIzvestiya Akodemii Nouk. Seriya Khimicheskaya, No. 11, pp. 2683–2688, November, 1996.