The maximum PI index of bicyclic graphs with even number of edges

Abstract The PI index of a graph G is defined by P I ( G ) = ∑ e = ( u , v ) ∈ E [ m u ( e | G ) + m v ( e | G ) ] where m u ( e | G ) be the number of edges in G lying closer to the vertex u than to the vertex v . In this paper, we give the upper bound on the PI index of connected bicyclic graphs with even number of edges and characterize the extremal graphs with maximum PI index.