On the soluble graph of a finite group

Let G be a finite insoluble group with soluble radical R(G). In this paper we
investigate the soluble graph of G, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in G\R(G), with x adjacent to y if they generate a soluble subgroup of G. Our main result states that this graph is always connected and its diameter, denoted δS (G), is at most 5. More precisely, we show that δS (G) 6 3 if G
is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that δS (Sn) = 3 for all n > 6. We also establish the existence of simple groups with δS (G) > 4. For instance, we prove that δS (A2p+1) > 4 for every Sophie Germain prime p > 5, which demonstrates that our general upper bound of 5 is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.