The hyper-Wiener index of graph operations

Authors:
M. H. Khalifeh;H. Yousefi-Azari;A. R. Ashrafi
Affiliations:
School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Islamic Republic of Iran;School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Islamic Republic of Iran;Department of Mathematics, Faculty of Science, University of Kashan, Kashan 87317-51167, Islamic Republic of Iran
Venue:
Computers & Mathematics with Applications
Year:
2008

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Abstract

Let G be a graph. The distance d(u,v) between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as WW(G)=12W(G)+12@?"{"u","v"}"@?"V"("G")d(u,v)^2. In this paper the hyper-Wiener indices of the Cartesian product, composition, join and disjunction of graphs are computed. We apply some of our results to compute the hyper-Wiener index of C"4 nanotubes, C"4 nanotori and q-multi-walled polyhex nanotori.