h-Polynomials of Graph Ideals via the Independence Polynomial

Edge and cover ideals are two standard ways to turn a graph $G$ into a squarefree monomial ideal, and their Hilbert series has a numerator called the h-polynomial. The theme of this talk is that a surprisingly large amount of this Hilbert-series data can be read directly from the independence polynomial $P_G(x)$, the generating function that counts independent sets of $G$. In I will explain how $P_G$ controls the h-polynomial, and in particular how the special evaluation $P_G(−1)$ (an alternating count of independent sets) and the multiplicity of the root $−1$ control the top coefficient and the degree of the numerator. I will illustrate these ideas on familiar graphs and on a simple “suspension” move that adds one new vertex and lets us track these invariants cleanly.