Geodesic
Wiener and vertex PI indices of the strong product of graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 749-769

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The Wiener index of a connected graph G, denoted by W(G), is defined as ½ ∑_u,v ∈ V(G)d_G(u,v). Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as ½W(G) + ¼ ∑_u,v ∈ V(G) d²_G(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, hyper-Wiener and vertex PI indices of the strong product G ⊠ K_m₀,m₁,...,m_r -1, where K_m₀,m₁,...,m_r -1 is the complete multipartite graph with partite sets of sizes m₀,m₁, ...,m_r -1, are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established.
Keywords: strong product, Wiener index, hyper-Wiener index, vertex PI index 
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     title = {Wiener and vertex {PI} indices of the strong product of graphs},
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Pattabiraman, K.; Paulraja, P. Wiener and vertex PI indices of the strong product of graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 749-769. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a10/