The Hilbert Series of the Face Ring of a Flag Complex

Abstract

It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of \(\), the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag complex in terms of those two invariants of \(\). Some computational issues are addressed and a recursive formula for the Hilbert series is given based on an algorithm of Bayer and Stillman.