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

TL;DR

This paper explores the relationship between the degree of the $h$-polynomial of a graph's edge ideal and the graph's independence number, providing combinatorial formulas for various graph classes.

Contribution

It establishes a necessary and sufficient condition linking the $h$-polynomial degree to the independence number using the independence polynomial at -1 and its derivatives.

Findings

Characterization of when $ ext{deg} ext{ } h_{R/I(G)}(t) = ext{independence number}$

Formulas for the $h$-polynomial degree for paths, cycles, bipartite, Cameron--Walker, and antiregular graphs

Use of independence polynomial derivatives at -1 to analyze $h$-polynomial degree

Abstract

Let $G=(V,E)$ be a finite simple graph. In this paper, we study the degree of the $h$-polynomial of the edge ideal of $G$ in relation to the independence number of $G$. Our approach is based on the value of the independence polynomial of $G$ at $-1$ and its derivatives at $-1$. We establish a necessary and sufficient condition for the equality $\deg\ h_{R/I(G)}(t)=\alpha(G)$. As consequences, we obtain combinatorial formulas for the degree of the $h$-polynomial for several classes of graphs, including paths, cycles, bipartite graphs, Cameron--Walker graphs, and antiregular graphs.