Weighted enumeration of spanning subgraphs with degree constraints

TL;DR

This paper develops a general method to prove Heilmann-Lieb type theorems for multivariate graph polynomials, which describe weighted spanning subgraphs with constraints, impacting probabilistic models and phase transition analysis.

Contribution

It introduces a broad framework for establishing nonvanishing regions of multivariate graph polynomials, extending Heilmann-Lieb theorems to new classes of spanning subgraph polynomials.

Findings

Established nonvanishing regions for new classes of graph polynomials

Extended Heilmann-Lieb theorems to multivariate generating functions

Implications for phase transition absence in probabilistic models

Abstract

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 $G$.