  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
$G$.
