  The generating graph encodes how generating pairs are spread among the
elements of a group. For more than ten years it has been conjectured that this
graph is connected for every finite group. In this paper, we give evidence
supporting this conjecture: we prove that it holds for all but a finite number
of almost simple groups and give a reduction to groups without non-trivial
soluble normal subgroups. Let $d(G)$ be the minimal cardinality of a generating
set for $G$. When $d(G)\geq3$, the generating graph is empty and the conjecture
is trivially true. We consider it in the more general setting of the rank
graph, which encodes how pairs of elements belonging to generating sets of
minimal cardinality spread among the elements of a group. It carries
information even when $d(G)\geq3$ and corresponds to the generating graph when
$d(G)=2$. We prove that it is connected whenever $d(G)\geq3$, giving tools and
ideas that may be used to address the original conjecture.
