A Kind of Non-commuting Graph of Finite Groups

Abstract

Let g be a fixed element of a finite group G. We introduce the g-noncommuting <br><br>graph of G whose vertex set is whole elements of the group G and two vertices x,y are <br><br>adjacent whenever [x,y] ≠ g and [y,x] ≠ g. We denote this graph by Γ _g, G. In this paper, <br><br>we present some graph theoretical properties of g-noncommuting graph. Specially, we <br><br>investigate about its planarity and regularity, its clique number and dominating number. <br><br>We prove that if G, H are isoclinic groups with |Z (G)|=|Z (H)|, then their associated <br><br>graphs are isomorphic.