Graph invariants in the spin model

  • Authors:

  • Alexander Schrijver

  • Affiliations:

  • CWI and University of Amsterdam, CWI, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2009

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Abstract

Given a symmetric nxn matrix A, we define, for any graph G,f"A(G):=@?@f:VG-{1,...,n}@?uv@?EGa"@f"("u")","@f"("v"). We characterize for which graph parameters f there is a complex matrix A with f=f"A, and similarly for real A. We show that f"A uniquely determines A, up to permuting rows and (simultaneously) columns. The proofs are based on the Nullstellensatz and some elementary invariant-theoretic techniques.