PI INDEX OF SOME BENZENOID GRAPHS

Abstract

The Padmakar-Ivan (PI) index of a graph G is defined as PI(G) = ∑(n eu (e|G)+ n ev (e|G)), where n eu (e|G) is the number of edges of G lying closer to u than to v, n ev (e|G) is the number of edges of G lying closer to v than to u and summation goes over all edges of G. In this paper, we first compute the PI index of a class of pericondensed benzenoid graphs consisting of n rows, n ≤ 3, of hexagons of various lengths. Finally, we prove that for any connected graph G with exactly m edges, PI(G) ≤ m(m-1) with equality if and only if G is an acyclic graph or a cycle of odd length. 1 . Here, we consider a new topological index, named the Padmakar-Ivan index, which is abbreviated as the PI index 2-17 . This newly proposed topological index, differ from the Wiener index 18 , the oldest topological index for acyclic (tree) molecules. We now describe some notations which will be adhered to throughout. Benzenoid systems (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 19 and in the references cited therein. Let G be a simple molecular graph without directed or multiple edges and without loops, the vertex and edge-shapes of which are represented by V(G) and E(G), respectively. The graph G is said to be connected if for every pair of vertices x and y in V(G) there exists a path between x and y. In this paper we only consider connected graphs. If e is an edge of G, connecting the vertices u and v then we write e=uv. The number of vertices of G is denoted by n. The distance between a pair of vertices u and w of G is denoted by d(u,w). We now define the PI index of a graph G. To do this, suppose that e = uv and introduce the quantities neu(e|G) and nev(e|G). neu(e|G) is the number of edges lying closer to vertex u than to vertex v, and nev(e|G) is the number of edges lying closer to vertex v than to vertex u. Then PI(G) = ∑(neu(e|G) + nev(e|G)), where the summation goes over all edges of G. Edges equidistant from both ends of the edge e = uv are not counted and the number of such edges is denoted by N(e). To clarify this, for every vertex u and any edge f = zw of graph G, we define d(f,u) = Min{d(u,w),d(u,z)}. Then f is equidistant from both ends of the edge e = uv if d(f,u) = d(f,v). In a series of papers, Khadikar and coauthors 2-17 defined and then computed the PI index of some chemical graphs. The present author 20 computed the PI index of a zig-zag polyhex nanotube. In this paper we continue this study to prove an important result concerning the PI index and find an exact expression for the PI index of some other chemical graphs. Our notation is standard and mainly taken from the literature. 21,22