On the Tutte polynomial of benzenoid chains

Abstract

The Tutte polynomial of a graph G, T(G, x,y) is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected. In this paper a simple formula for computing Tutte polynomial of a benzenoid chain is presented. 1. I NTRODUCTION Benzenoid graphs or graph representations of benzenoid hydrocarbons are defined as finite connected plane graphs with no cut-vertices, in which all interior regions are mutually congruent regular hexagons. More details on this important class of molecular graphs can be found in the book of Gutman and Cyvin (1), and in the references cited therein. Suppose G is an undirected graph, E = E(G) and v is a vertex of G. The vertex v is reachable from another vertex u if there is a path in G connecting u and v. In this case we write vu. A single vertex is a path of length zero and so  is reflexive. Moreover, we can easily prove that  is symmetric and transitive. So  is an equivalence relation on V(G). The equivalence classes of  is called the connected components of G. The Tutte polynomial of a graph G is a polynomial in two variables defined for every undirected graph contains information about how the graph is connected (2-4). To define we need some notions. The edge contraction G/uv of the graph G is the graph obtained by merging the vertices u and v and removing the edge uv. We write G − uv for the graph where the edge uv is merely removed. Then the Tutte polynomial of G is defined by the recurrence relation T(G; x, y) = T(G  e; x, y) + T(G/e; x, y) if e is neither a loop nor a bridge with base case T(G; x, y) = x i y j if G contains i bridges and j loops and no other edges. In particular,