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2011•Discrete Mathematics Algorithms and ApplicationsRequires access

THE HYPER-WIENER INDEX OF THE k th POWER OF A GRAPH

Weijuan Zhang, Baoyindureng Wu, Xinhui An

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Abstract

The k th power of a graph G, denoted by Gk, is a graph whose vertex set is V(G), two distinct vertices being adjacent in Gk if and only if their distance in G is at most k. The hyper-Wiener index WW(G) of a graph G is defined as [Formula: see text], where dG(u,v) is the distance between vertices u and v in G. In this paper, the bounds on the hyper-Wiener index of the graph Gk are given. The Nordhaus–Gaddum-type inequality for the hyper-Wiener-index of the graph Gk is also presented.

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What this paper is about

The k th power of a graph G, denoted by Gk, is a graph whose vertex set is V(G), two distinct vertices being adjacent in Gk if and only if their distance in G is at most k. The hyper-Wiener index WW(G) of a graph G is defined as [Formula: see text], where dG(u,v) is the distance between vertices u and v in G. In this paper, the bounds on the hyper-Wiener index of the graph Gk are given. The Nordhaus–Gaddum-type inequality for the hyper-Wiener-index of the graph Gk is also presented.

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Available abstract

The k th power of a graph G, denoted by Gk, is a graph whose vertex set is V(G), two distinct vertices being adjacent in Gk if and only if their distance in G is at most k. The hyper-Wiener index WW(G) of a graph G is defined as [Formula: see text], where dG(u,v) is the distance between vertices u and v in G. In this paper, the bounds on the hyper-Wiener index of the graph Gk are given. The Nordhaus–Gaddum-type inequality for the hyper-Wiener-index of the graph Gk is also presented.

Key concepts: Wiener index, Combinatorics, Mathematics, Graph, Vertex (graph theory), Bound graph, Distance-regular graph, Discrete mathematics

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