Groups with the same non-commuting graph

Abstract

The non-commuting graph Γ G of a non-abelian group G is defined as follows. The vertex set of Γ G is G − Z ( G ) where Z ( G ) denotes the center of G and two vertices x and y are adjacent if and only if x y ≠ y x . It has been conjectured that if G and H are two non-abelian finite groups such that Γ G ≅ Γ H , then | G | = | H | and moreover in the case that H is a simple group this implies G ≅ H . In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H . Then for certain simple groups H , we show that the graph isomorphism Γ G ≅ Γ H implies G ≅ H .