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 SG (x, y), 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 G. Some computational issues are addressed and a recursive formula for the Hilbert series is given based on an algorithm of Bayer and Stillman.

• Type: Journal
• Year: 2002
• Where: GC
• Authors: Paul Renteln
• Keywords: Applied Computing, Flag Complex, GC 2002, Hilbert Series, Subgraph Polynomial