Bounds On The Weighted Vertex Pi Index Of Cacti Graphs

The weighted vertex PI index of a graph G is defined byPIw(G) = Sigma(e=uv is an element of E(G)) (d(G)(u)+d(G)(v))(n(u)(e vertical bar G) + n(v)(e vertical bar G)where d(G)(u) denotes the vertex degree of u and n(u)(e vertical bar G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.