Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutions

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.