Group nilpotency from a graph point of view (Q6618056)

This paper gives a quite comprehensive survey of the literature regarding the graphs \(\Gamma_{G}\) which can be associated with a group \(G\). In particular, they focus on the problem of understanding whether or not a finite group \(H\) must be nilpotent provided \(\Gamma_{H}\) is isomorphic to \(\Gamma_{G}\) for a finite nilpotent group \(G\).\N\NThe authors bring a new contribution to the issue mentioned above by proving Theorem 5.1: Let \(G\) and \(H\) be finite \(2\)-generated groups with isomorphic generating graphs. If \(G\) is nilpotent and \(H\) is supersoluble, then \(H\) is nilpotent.