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

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 $\mathrm{°}{h}_{R\mathrm{/}I(G)}(t)=\alpha (G)$. 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.