The automorphism group of commuting graph of a finite group. (Q2877818)

Let \(G\) be a finite group. The commuting graph \(\Delta(G)\) of \(G\) is a graph whose vertex set is \(G\) and two vertices \(x\) and \(y\) are adjacent if and only if \(x\neq y\) and \(xy=yx\). In this paper, the authors study the automorphism group of the graph \(\Delta(G)\). It is proved that \(\Aut(\Delta(G))\) is abelian or primary if and only if \(|G|\leq 2\), and \(\Aut(\Delta(G))\) is square-free if and only if \(|G|\leq 3\). Some new graphs that are useful in studying the group \(\Aut(\Delta(G))\) are presented and their main properties are investigated.