Nordhaus-Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts

  • Authors:

  • Guifu Su;Liming Xiong;Yi Sun;Daobin Li

  • Affiliations:

  • School of Mathematics, Beijing Institute of Technology, Beijing, 100081, PR China;School of Mathematics, Beijing Institute of Technology, Beijing, 100081, PR China and Department of Mathematics, Jiangxi University, Jiangxi, 330026, PR China;College of Mathematics and System Science, Xinjiang University, Xinjiang, 830046, PR China;School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing, 100083, PR China

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

Let k=2 be an integer, a k-decomposition(G"1,G"2,...,G"k) of a graph G is a partition of its edge set to form k spanning subgraphs G"1,G"2,...,G"k. The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW=WW(G):=12W(G)+12W"2(G), where W is the Wiener index (Wiener 1947 [18]) and W"2 is the sum of squares of distance of all pairs of vertices in G. In this paper, we investigate the Nordhaus-Gaddum-type inequality of a 3-decomposition of K"n for the hyper-Wiener index: 7n2@?WW(G"1)+WW(G"2)+WW(G"3)@?2n+24+n2+4(n-1). The corresponding extremal graphs are characterized.