The Wiener index of a connected graph G, denoted by W (G), is defined asThe 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, hyper-Wiener and vertex PI indices of the strong product G ⊠ K m0,m1,...,mr−1 , where K m0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m 0 , m 1 , . . . , m r−1 , are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established.
In this paper, we present exact formula for the weighted PI index of corona product of two connected graphs in terms of other graph invariants including the PI index, first Zagreb index and second Zagreb index. Then, we apply our result to compute the weighted PI indices of t-fold bristled graph, bottleneck graph, sunlet graph, star graph, fan graph, wheel graph and some classes of bridge graphs.
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