The Automorphism Group of Commuting Graph of a Finite Group

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and <TEX>x,y∈X</TEX> (<TEX>x≠y</TEX>) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by <TEX>Δ(G)</TEX>. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(<TEX>Δ(G)</TEX>) is abelian if and only if <TEX>G≤2</TEX>; <TEX>Aut(Δ(G))</TEX> is of prime power if and only if <TEX>G≤2</TEX>, and <TEX>Aut(Δ(G))</TEX> is square-free if and only if <TEX>G≤3</TEX>. Some new graphs that are useful in studying the automorphism group of <TEX>Δ(G)</TEX> are presented and their main properties are investigated.