A benzenoid is a class of chemical compounds with at least one benzene ring(hexagon as a graph) and resonance bonds in the benzene ring give increased stability in benzenoids. A finite connected ...subgraph of the infinite hexagonal lattice without cut vertices or non-hexagonal interior faces is said to be a benzenoid system (or a hexagonal system) and these systems are geometric figures. Benzenoid systems are widely used because they are the representations of the skeletons (focussing on the structure induced by the carbon atoms) of molecules of benzenoid hydrocarbons.
The first and second Zagreb indices are among the most studied topological indices. We now consider analogous graph invariants, based on the second degrees of vertices, called Zagreb connection indices. The main objective of this paper is to compute these connection indices for six benzenoid systems(three catacondensed and three pericondensed systems). In the end, an application of the obtained results for these indices of some classes of benzenoid systems is also included. Mainly, a comparison among the Zagreb connection indices of some benzenoid systems is performed with the help of numerical tables and 3 D plots.
Let G=(V,E) be a molecular graph, in which the vertices of V represent atoms and the edges of E the bonds between pairs of atoms. One of the earliest and most widely studied degree-based molecular ...descriptor, the Randić index of G, is defined as R(G)=∑uv∈E(dudv)−12, where du denotes the degree of u ∈ V. Bollobás and Erdős (1998) generalized this index by replacing −12 with any fixed real number. To facilitate the enumeration of these indices, motivated by earlier work of Dvořák et al. (2011) and Knor et al. (2015) introduced two other indices, that provide lower and upper bounds, respectively: Rα′(G)=∑uv∈Emin{duα,dvα} and Rα″(G)=∑uv∈Emax{duα,dvα}. In this paper, we give expressions for computing Rα′ and Rα″ of benzenoid systems and phenylenes, as well as a relation between Rα′ and Rα″ of a phenylene and its corresponding hexagonal squeeze. We also determine the extremal values of Rα′ and Rα″ in benzenoid systems with h hexagons for different intervals for the value of α.
A catacondensed benzenoid system (resp. benzenoid chain) is a benzenoid system whose inner dual graph is a tree (resp. a path). The Tutte polynomial of a graph is a two-variable polynomial whose ...evaluations at various points are equivalent to the exact solutions of many counting problems. In this paper, we introduce a graph vector at a given edge which related to the Tutte polynomial. Based on this concept and by three classes transfer matrices, we get the reduction formula for Tutte polynomial of any catacondensed benzenoid system. Moreover, the number of spanning trees for any catacondensed benzenoid system is also determined via a product of
(
2
×
2
)
matrices with entries in
N
. As a by-product, we study the extremum problem of the number of spanning trees over the set of cataconsed hexagonal systems with one branched hexagon.
The Fibonacci dimension fdim(G) of a graph G was introduced in Cabello et al. (2011) 1 as the smallest integer d such that G admits an isometric embedding into Γd, the d-dimensional Fibonacci cube. ...The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacondensed benzenoid systems. Our results show that the Fibonacci dimension of the resonance graph of a catacondensed benzenoid system G depends on the inner dual of G. Moreover, we show that computing the Fibonacci dimension can be done in linear time for a graph of this class.
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