Realizable (reg, deg h)-Pairs for Cover Ideals via Independence Polynomials

Let $G$ be a finite simple graph on $n$ vertices and set $R=\Bbbk[x_1,\dots,x_n]$, with edge ideal $I(G)$ and cover ideal $J(G)$. We give an explicit description of the $h$-polynomial of $R/J(G)$, in a form that extends to the Alexander dual of any squarefree monomial ideal. We then express $\textrm{deg } h_{R/I(G)}(t)$ and $\textrm{deg } h_{R/J(G)}(t)$ in terms of the independence polynomial $P_G(x)=\sum_{i\ge 0} g_i x^i$ via an invariant $M(G)$, the multiplicity of $x=-1$ as a root of $P_G(x)$. In particular, we prove [\textrm{deg } h_{R/I(G)}(t)=α(G)-M(G) \qquad\text{and}\qquad \textrm{deg } h_{R/J(G)}(t)=n-2-M(G), ] where $α(G)$ is the independence number of $G$. As a corollary, $M(G)$ is the additive inverse of the $\mathfrak{a}$-invariants of $R/I(G)$ and $R/J(G)$. We develop recursions and closed formulas for $M(G)$ for broad graph families, and use them to analyze which (reg, deg h)-pairs occur for cover ideals within chordal classes, including explicit constructions realizing extremal behavior. We conclude with a conjectural bound on $\left|\textrm{reg }(R/J(G))-\textrm{deg } h_{R/J(G)}(t)\right|$ for connected graphs.

💡 Research Summary

The paper investigates the interplay between two fundamental invariants of the cover ideal J(G) of a finite simple graph G on n vertices: the Castelnuovo‑Mumford regularity reg (R/J(G)) and the degree of its h‑polynomial, deg h_{R/J(G)}(t). The authors introduce a new graph invariant M(G), defined as the multiplicity of the root x = −1 of the independence polynomial P_G(x)=∑_{i≥0} g_i x^i (where g_i counts independent sets of size i). They prove that M(G) governs both the a‑invariant of R/J(G) and the degree of its h‑polynomial:

 deg h_{R/J(G)}(t) = n − 2 − M(G) and a(R/J(G)) = −M(G).

This result mirrors a known formula for edge ideals I(G) and follows from an explicit expression for the h‑polynomial of R/J(G) (Theorem 3.5). By Alexander duality, the a‑invariant of R/J(G) equals that of R/I(G), so the same M(G) appears in both contexts.

To make M(G computable, the authors develop recursive formulas and closed forms for several graph families. For trees of radius at most two, M(G) is either 0 or 1, leading to deg h = n−2 or n−α(G)−1 respectively. For split graphs (graphs whose vertex set splits into a clique C and an independent set I), they obtain a closed formula P_G(x)= (1+x)^{|C|}·∑_{j=0}^{|I|} \binom{|I|}{j} x^j, which yields M(G)=α(G)−|C|. Consequently every connected split graph lies on the diagonal reg = deg h, and the possible regularities are precisely n−|C|−1.

In the chordal setting the paper constructs two extremal families. The first family attains the maximal regularity reg = n−2 while allowing deg h to range from n−α(G)−1 up to n−2, producing vertical “strings” in the (reg,deg h) plane. The second family lies on the line deg h = reg−1, showing that chordal graphs can realize the lower bound of the difference. By contrast, for block graphs the authors prove deg h ≥ reg−1, so this class never falls below the diagonal.

Extensive computer experiments for all connected graphs up to n≈10 vertices are reported. The data suggest a universal bound on the absolute difference between regularity and h‑polynomial degree:

 |reg(R/J(G)) − deg h_{R/J(G)}(t)| ≤ ⌊log₂ n⌋ − 1,

which is stated as Conjecture 12.4. The conjecture holds for all tested graphs and aligns with the theoretical extremal constructions presented.

Overall, the work provides a clean combinatorial bridge—via the root multiplicity M(G) of the independence polynomial—between homological invariants of cover ideals and classical graph parameters such as independence number and domination number. It delivers explicit formulas, recursive tools, and extremal constructions that together map out the feasible region of (reg,deg h) pairs for cover ideals, especially within chordal and split graph families. This deepens our understanding of how graph structure controls algebraic complexity and opens avenues for further exploration of the proposed bound.