Commuting involution graphs for symmetric groups.
Let \(G\) be a finite group and \(X\) be a subset of \(G\). The commuting graph \({\mathcal C}(G,X)\) on the set \(X\) has \(X\) as its vertex set with \(x,y\in X\) joined by an edge whenever \(xy=yx\). If \(X\) consists entirely of involutions, then \({\mathcal C}(G,X)\) is called a commuting involution graph. Many authors have studied \({\mathcal C}(G,X)\) for different choices of \(G\) and \(X\), and from a number of different perspectives. In this paper, the authors analyse the properties of the graph \({\mathcal C}(G,X)\) where \(G\) is the symmetric group \(\text{Sym}(n)\) of degree \(n\) and \(X=a^G\) is an involution conjugacy class of \(G\). This analysis is primarily focused upon the structure and sizes of the sets \(\Delta_i(a)\), the \(i\)-th disc of \(a\) (\(i\in\mathbb{N}\)), connected components of \({\mathcal C}(G,X)\) and, when \({\mathcal C}(G,X)\) is connected, the diameter \(\text{Diam\,}{\mathcal C}(G,X)\) of \({\mathcal C}(G,X)\). We may suppose, without loss of generality, that \(a=(12)(34)\dots(2m-1\,2m)\). It is proved that \({\mathcal C}(G,X)\) is disconnected if and only if \(n=2m+1\) or \(n=4\) and \(m=1\). If \(n=4\) and \(m=1\), then \({\mathcal C}(G,X)\) consists of three connected components each of size 2. If \(n=2m+1\), then \({\mathcal C}(G,X)\) has \(n\) connected components each of which is isomorphic to \({\mathcal C}(H,Y)\) where \(H\cong\text{Sym}(2m)\), \(Y=b^H\), and \(b=(12)(34)\dots(2m-1\,2m)\). Using various results concerning \(\Delta_i(a)\), the authors prove also that if \({\mathcal C}(G,X)\) is connected then either \(\text{Diam\,}{\mathcal C}(G,X)\leq 3\), or \(2m+2\in\{6,8,10\}\) and \(\text{Diam\,}{\mathcal C}(G,X)=4\).
scientific article; zbMATH DE number 1967903
scientific article