G. Clar 6n Rule versus Hückel 4n + 2 Rule 3464 H. Hydrocarbons versus Heteroatomic Systems 3465 IV. Hidden Treasures of Kekulé Valence Structures 3466 A. Conjugated Circuits 3467 B. Innate Degree of Freedom 3470 C. Clar Structures 3472 V. Graph Theoretical Approach to Chemical Structure 3473 A. Metric 3473 B. Chemical Graphs 3473 C. Isospectral Graphs 3473 D. Embedded Graphs 3475 E. Partial Ord...

In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physico-chemical properties of molecules. As the main result, we de...

The Schultz molecular topological index (MTI) is compared with the Wiener index (W) of a molecular graph. It is shown that 2VminW < MTI e 4VmaxW holds for any (connected) graph Γ, with Vmin and Vmax denoting the smallest and the largest valency, respectively, of the vertices of Γ. For molecular graphs these bounds can be further improved. For instance, for benzenoid systems we have the followin...

A matching M of a graph G is maximal if it not proper subset any other in G. Maximal matchings are much less known and researched than their maximum perfect counterparts. In this paper we present the recurrences generating functions for sequences enumerating two classes chemically interesting linear polymers: polyspiro chains benzenoid chains. We also analyze asymptotic behavior those determine...

Hexagonal chains are a special class of catacondensed benzenoid system and phenylene polycyclic aromatic compounds. Recently, A family Sombor indices was introduced by Gutman in the chemical graph theory. It had been examined that these may be successfully applied on modeling thermodynamic properties In this paper, we study expected values random hexagonal chains, consider some graphs such as g...

The geometric-arithmetic index of graph G is defined as GA(G) = ∑ uv∈E(G) 2 √ dudv du+dv , du (or dv) is the degree the vertex u (or v). The GA index of benzenoid systems and phenylenes are computed, a simple relation is established between the geometric-arithmetic of a phenylene and the corresponding hexagonal squeeze in this paper. Mathematics Subject Classification: 05C05, 05C12

Let G be a planar embedded 2-connected graph. Then the vertices of its resonance graph R(G) are the 1-factors of G, two 1-factors being adjacent whenever their symmetric difference is a bounded face of G. For a class of graphs containing the chemically important catacondensed benzenoid graphs we show that the resonance graphs are median. In particular, if G belongs to this class, R(G) has an is...

Among the variety of molecular-graph-based structure-descriptors, aimed at measuring the extent of branching of the skeleton of organic molecules, there are the Wiener index W and the spectral radius λ1 i. e., the greatest eigenvalue of the molecular graph. Although, these indices are used for decades, they have never been compared. Our computational studies show that within series of the isome...

A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the study of benzenoid hydrocarbons and fullerenes in computati...

The number of resonance structures (SC) for previously published concealed non-Kekuléan benzenoid hydrocarbons is determined. Using a simple computer program, analytical expressions for determining SC for various classes of non-Kekuléan (free-radical) benzenoid hydrocarbons are derived, and some properties of concealed non-Kekuléan benzenoid hydrocarbons are studied.