  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 $G \setminus R(G)$, 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 $\delta_{\mathcal{S}}(G)$, is at most $5$.
More precisely, we show that $\delta_{\mathcal{S}}(G) \leqslant 3$ if $G$ is
not almost simple and we obtain stronger bounds for various families of almost
simple groups. For example, we will show that $\delta_{\mathcal{S}}(S_n) = 3$
for all $n \geqslant 6$. We also establish the existence of simple groups with
$\delta_{\mathcal{S}}(G) \geqslant 4$. For instance, we prove that
$\delta_{\mathcal{S}}(A_{2p+1}) \geqslant 4$ for every Sophie Germain prime $p
\geqslant 5$, 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.
