Discussiones Mathematicae Graph Theory 32(4) (2012) 749-769
doi: 10.7151/dmgt.1647

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Wiener and Vertex PI Indices of the Strong Product of Graphs

K. Pattabiraman and P. Paulraja

Department of Mathematics, Annamalai University
Annamalainagar 608 002, India

Abstract

The Wiener index of a connected graph G, denoted by W(G), is defined as ½ ∑u,v ∈ V(G)dG(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)d2G(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⊠ 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.

Keywords: strong product, Wiener index, hyper-Wiener index, vertex PI index

2010 Mathematics Subject Classification: 05C12, 05C76.

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Received 20 June 2011
Revised 25 January 2012
Accepted 27 January 2012