First and second extremal bipartite graphs with respect to PI index

The Padmakar–Ivan (PI) index of a graph GG is defined as the sum of terms [mu(e)+mv(e)mu(e)+mv(e)] over all edges of G, where ee is an edge, connecting the vertices uu and vv, where mu(e)mu(e) is the number of edges of GG lying closer to the vertex uu than to the vertex vv, and where mv(e)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.