  In this note we define and study graph invariants generalizing to higher
dimension the maximum degree of a vertex and the vertex-connectivity (our
$0$-dimensional cases). These are known to coincide almost surely in any regime
for Erdoes-Renyi random graphs. We show the same in the one dimensional case
for a middle density regime and show the easier inequality for all dimensions
in the same regime. Our original motivation comes from the study of minimal
free resolutions of Stanley-Reisner rings of clique complexes of graphs in
commutative algebra, In that setting the higher dimensional vertex
connectivities determine the lengths of the strands. Through our results we aim
to replace (asymptotically) vertex connectivity by a simpler invariant.
