First and second extremal bipartite graphs with respect to PI index

Abstract

The Padmakar-Ivan (PI) index of a graph G is defined as the sum of terms [mu(e)+mv(e)] over all edges of G, where e is an edge, connecting the vertices u and v, where mu(e) is the number of edges of G lying closer to the vertex u than to the vertex v, and where mv(e) is defined analogously. The extremal values of the PI index are determined in the class of connected bipartite graphs with a given number of edges. © 2011 Elsevier Ltd.