The weighted vertex PI index of bicyclic graphs

The weighted vertex PI index of a graph G is defined by PIw(G)=∑e=uv∈E(G)(dG(u)+dG(v))(nu(e|G)+nv(e|G))where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. In this paper, we give the upper and lower bounds on the weighted vertex PI index of connected bicyclic graphs and completely characterize the corresponding extremal graphs.