The Harary index of the non-commuting graph for dihedral groups

Abstract

Assume G is a non-abelian group which consists a set of vertices, V = {v1 , v2, ..., vn} and a set of edges, E = {e1, e2, ..., em} where n and m are the positive integers. The non-commuting graph of G, denoted by ΓG, is the graph of vertex set G−Z(G), whose vertices are non-central elements, in which Z(G) is the center of G and two distinct vertices are adjacent if and only if they do not commute. In addition, the Harary index of a graph ΓG is the half-sum of the elements in the reciprocal distance of Dij where Dij the distance between vertex i and vertex j. In this paper, the Harary index of the non-commuting graph for dihedral groups is determined and its general formula is developed.