Independence Polynomials of graphs and degree of $h$-polynomials of edge ideals

TL;DR

This work links the degree of the $h$-polynomial of a graph edge ideal to the independence polynomial. By expressing the Hilbert series in terms of $I(G,t)$ and its derivatives at $-1$, it gives a sharp criterion: $\deg h_{R/I(G)}(t)=\alpha(G)$ if and only if $I(G,-1)\neq 0$, with a precise reduction when $I(G,-1)=0$. The authors derive explicit, combinatorial formulas for the degree in several graph families (paths, cycles, bipartite graphs, Cameron–Walker graphs, and antiregular graphs) and present a general reduction framework that clarifies when the maximum degree is attained. The results unify and simplify known formulas, offer elementary proofs for certain classes, and provide practical criteria for computing $\deg h_{R/I(G)}(t)$ from graph structure.

Abstract

Let $G = (V, E)$ be a finite simple graph. In this paper, we characterize the degree of the $h$-polynomial of the edge ideal of $G$ in terms of the independence number of $G$. The key tools are the value of the independence polynomial of $G$ at $-1$ and its derivative. Using this approach, we obtain, in particular, combinatorial formulas for the degree of the $h$-polynomial of paths, cycles, bipartite graphs, Cameron-Walker graphs and antiregular graphs.

Theorems & Definitions (61)

• Theorem 1
• Lemma 1.1: Hoede, Theorem 2.3
• Lemma 1.2: Hoede, Corollary 3.8 and Theorem 3.9
• Lemma 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 3.1