Vertex and edge PI indices of Cartesian product graphs

The Padmakar–Ivan (PI) index of a graph G   is the sum over all edges uvuv of G   of the number of edges which are not equidistant from uu and vv. 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.