On some topological indices of the tensor products of graphs

The Wiener index   of a connected graph GG, denoted by W(G)W(G), is defined as 12∑u,v∈V(G)dG(u,v). Similarly, hyper-Wiener index   of a connected graph GG, denoted by WW(G)WW(G), is defined as 12W(G)+14∑u,v∈V(G)dG2(u,v). The Padmakar–Ivan (PI) index of a graph GG is the sum over all edges uvuv of GG of the number of vertices which are not equidistant from uu and vv. In this paper, we obtain the exact formulas for Wiener, the hyper-Wiener and PI indices of the tensor 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. Using the results obtained here, the main theorems proved in Hoji et al. (2010) [11] are obtained as corollaries. Also we have obtained lower bounds for Wiener and hyper-Wiener indices of tensor products of graphs.