Weighted enumeration of spanning subgraphs with degree constraints

Details Weighted enumeration of spanning subgraphs with degree constraints Weighted enumeration of spanning subgraphs with degree constraints

2008-03-11

arxiv.org/abs/0803.1659v1

Mathematics

Combinatorics

complete re-write of arXiv:math/0412059 with some new results

Scientific paper

The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial $\sum_k m_k(G) y^k$ has only real nonpositive zeros, in which $m_k(G)$ is the number of $k$-edge matchings of a graph $G$. There is a stronger multivariate version of this theorem. We provide a general method by which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph $G$. These polynomials are multivariate generating functions for spanning subgraphs of $G$ with certain weights and constraints imposed, and the theorems specify regions in which these polynomials are nonvanishing. Such theorems have consequences for the absence of phase transitions in certain probabilistic models for spanning subgraphs of $G$.

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