Volkmar Welker: Higher dimensional vertex connectivity of random simplicial complexes

(joint work with Eric Babson)

Motivated by questions in commutative algebra we study a higher dimensional analog of the vertex connectivity of a graph in a probabilistic setting. We study the clique complex of an Erdoes-Renyi graph; i.e., the simplicial complex of all vertex sets of cliques. Higher vertex connectivity is defined as a homological invariant of the complex and depends on the coefficient ring. We conjecture a higher dimensional analog of the fact that vertex connectivity and minimal vertex degree coincide aas. We verify one inequality in all dimensions and for all coefficient rings and prove equality in dimension 1 in a middle density setting.