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arXiv · 2602.12169

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

Ton That Quoc Tan
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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.

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Ton That Quoc Tan. 2026-02-12. Independence Polynomials of graphs and degree of $h$-polynomials of edge ideals. https://arxiv.org/abs/2602.12169

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