The Hilbert Series of the Face Ring of a Flag Complex

 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. Received: December 10, 1999 Acknowledgments. I would like to thank Rick Wilson and the mathematics department of the California Institute of Technology for their kind hospitality, and Richard Stanley for pointing out an error in an earlier draft.