Vertex and edge PI indices of Cartesian product graphs

Abstract

The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number of edges which are not equidistant from u and v. In this paper, the notion of vertex PI index of a graph is introduced. We apply this notion to compute an exact expression for the PI index of Cartesian product of graphs. This extends a result by Klavzar [On the PI index: PI-partitions and Cartesian product graphs, MATCH Commun. Math. Comput. Chem. 57 (2007) 573-586] for bipartite graphs. Some important properties of vertex PI index are also investigated.

Wiener-type invariants of some graph operations

“…In [14], Klavžar, Rajapakse and Gutman computed the Szeged index of the Cartesian product graphs. The present authors, [8,9,10,11,12,13,22], computed some exact formulae for the hyper-Wiener, vertex PI, edge PI, the first Zagreb, the second Zagreb, the edge Wiener and the edge Szeged indices of some graph operations. The aim of this section is to continue this program for computing the Wiener-type invariants for five graph operations.…”

GA2 index of some graph operations

“…In [17], Klavžar, Rajapakse and Gutman computed the Szeged index of the Cartesian product graphs. The recent authors, [1,2,9,11,12,13,14,15,16,18,24], computed some exact formulas for the hyper-Wiener, vertex PI, edge PI, the first Zagreb, the second Zagreb, the edge Wiener and the edge Szeged indices of some graph operations. The aim of this section is to continue this program for computing the GA 2 index of these graph operations.…”

PI polynomials of product graphs

“…In [11], Khalifeh, Yousefi-Azari and Ashrafi computed the PI index of Cartesian product graphs. Here we continue this progress by computing the PI polynomial of Cartesian product graphs.…”