The Hilbert Series of the Face Ring of a Flag Complex | Zendy

Paul Renteln | Zendy

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The Hilbert Series of the Face Ring of a Flag Complex

Author(s) -

Paul Renteln

Publication year - 2002

Publication title -

graphs and combinatorics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.59

H-Index - 40

eISSN - 1435-5914

pISSN - 0911-0119

DOI - 10.1007/s003730200045

Subject(s) - flag (linear algebra) , mathematics , series (stratigraphy) , hilbert–poincaré series , ring (chemistry) , combinatorics , face (sociological concept) , pure mathematics , discrete mathematics , algebra over a field , linguistics , paleontology , biology , chemistry , philosophy , organic chemistry

It is shown that the Hilbert series of the face ring of a clique complex (equiva- lently, 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.W ealso find a simple relation- ship 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.

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