On the PI polynomial of a graph

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 define the PI polynomial of a graph and investigate some of the elementary properties of this polynomial and compute it for some well-known graphs. Finally, we generalize some of the properties of Wiener polynomial to PI polynomial.