Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

Jonathan D Leake,Nick R Ryder

doi:10.5802/alco.63

Abstract

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.

Highlights

Given a graph G = (V, E), the matching polynomial of G and the independence polynomial of G are defined as follows. μ(G) := −x2 |M| I(G) := x|S|
Realrootedness implies log-concavity and unimodality of the matchings of a graph, and recently in [15] the root bound was used to show the existence of Ramanujan graphs
It is well-known that the matching polynomial of a graph G is equal to the independence polynomial of the line graph of G

Summary

Introduction

The real-rootedness of the matching polynomial and the Heilmann–Lieb root bound are important results in the theory of undirected simple graphs. The first part of this paper is a partial generalization of this stability result to the multivariate independence polynomial of clawfree graphs. We prove a result related to the real-rootedness of certain weighted independence polynomials. This result was originally proven by Engström in [9] by bootstrapping the Chudnovsky and Seymour result for rational weights and using density arguments. We prove the Heilmann–Lieb root bound for the independence polynomial of a certain subclass of claw-free graphs. By considering a particular graph called the Schläfli graph, we demonstrate that this root bound does not hold for all claw-free graphs and provide a weaker bound in the general claw-free case

Stability Theory

Multivariate Graph Polynomials and Stability

Root Bounds

Failure of the Root Bounds

Other Remarks