Wiener and vertex PI indices of the strong product of graphs

The Wiener index of a connected graph G, denoted by W(G), is defined as 1 P u,v∈V (G) dG(u,v). Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as 1 W(G) + 1 P u,v∈V (G) d 2 (u,v). The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyperWiener and vertex PI indices of the strong product G ⊠ Km0,m1,...,mr−1 , where Km0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1,...,mr−1, are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established.