On Wiener index of graph complements

Document Type: Research Paper

Authors

1 Anna University

2 University of Kragujevac Kragujevac, Serbia

Abstract

‎Let $G$ be an $(n,m)$-graph‎. ‎We say that $G$ has property $(\ast)$‎ ‎if for every pair of its adjacent vertices $x$ and $y$‎, ‎there ‎exists a vertex $z$‎, ‎such that $z$ is not adjacent‎ ‎to either $x$ or $y$‎. ‎If the graph $G$ has property $(\ast)$‎, ‎then‎ ‎its complement $\overline G$ is connected‎, ‎has diameter 2‎, ‎and its‎ ‎Wiener index is equal to $\binom{n}{2}+m$‎, ‎i.e.‎, ‎the Wiener index‎ ‎is insensitive of any other structural details of the graph $G$‎. ‎We characterize numerous classes of graphs possessing property $(\ast)$‎, ‎among which are trees‎, ‎regular‎, ‎and unicyclic graphs‎.

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