Algebraic Invariants of Edge Ideals Under Suspension

The central question of this paper is: how do algebraic invariants of edge ideals change under natural graph operations? We study this question through the lens of suspensions. The (full) suspension of a graph is obtained by adjoining a new vertex adjacent to every vertex of the original graph; this construction is well-understood in the literature. Motivated by the fact that regularity is preserved under full suspension while projective dimension becomes maximal, we refine the construction to selective suspensions, where the new vertex is joined only to a prescribed subset of vertices. We focus on two extremal choices: minimal vertex covers and maximal independent sets. For suspensions over minimal vertex covers of an arbitrary graph, regularity is preserved and projective dimension increases by one. Moreover, the independence polynomial changes in a controlled way, allowing us to track $\mathfrak a$-invariants under cover suspension. In contrast, the analogous uniform behavior fails in general for suspensions over maximal independent sets. We therefore analyze paths and cycles and give a complete description: projective dimension always increases by one, and regularity and the $\mathfrak a$-invariant are preserved except for a unique extremal family of paths, where both invariants increase by one.

💡 Research Summary

This paper presents a systematic investigation into how fundamental algebraic invariants of edge ideals—Castelnuovo-Mumford regularity, projective dimension, and the a-invariant—behave under a natural graph operation called “selective suspension.”

The study is situated within combinatorial commutative algebra, where a finite simple graph G is associated with its edge ideal I(G), generated by monomials x_i x_j corresponding to edges. The homological properties of the quotient ring R/I(G), encoded in its minimal free resolution, are deeply linked to the topology of the graph’s independence complex Δ(G). A central theme is understanding how modifying the graph translates into predictable changes in these algebraic invariants.

The authors focus on “suspension” operations. The classical full suspension ̂G adds a new vertex z adjacent to every vertex of G. It is known that regularity is preserved under this operation, while projective dimension becomes maximal. Motivated by this, the authors introduce and analyze selective suspensions. Given a subset C of vertices of G, the C-suspension G(C) is formed by adding z and connecting it precisely to all vertices in C. The corresponding edge ideal is I(G(C)) = I(G) + (z*x_i : x_i ∈ C).

The research concentrates on two extremal and complementary choices for C: minimal vertex covers and maximal independent sets. The main findings reveal a striking dichotomy in behavior.

For suspensions over a minimal vertex cover, the effect is uniform across all graphs. The authors prove that regularity is preserved, and projective dimension increases by exactly one (Theorem 3.5). Furthermore, the independence polynomial transforms in a controlled manner: P_{G(C)}(x) = P_G(x) + x * P_{G-C}(x). This allows for tracking the a-invariant via the multiplicity M(G) of x=-1 as a root of the independence polynomial (Proposition 3.6).

In contrast, suspensions over a maximal independent set do not exhibit such uniform behavior in general. To pinpoint the underlying mechanics, the authors provide a complete analysis for two fundamental graph families: cycles and paths.

For cycles C_n, the behavior turns out to be uniform again: regardless of the chosen maximal independent set C, projective dimension increases by one, while regularity and the a-invariant are preserved (Theorems 4.7 and 4.9). The most technically involved case occurs when the cycle length n is a multiple of 3 and C has size ⌈n/3⌉ (the “wide-spoke” configuration). This case is resolved by constructing an explicit acyclic matching on the independence complex Δ(G(C)) using discrete Morse theory, which shows the vanishing of the top homology and thus confirms the regularity preservation.

For paths P_n, a more nuanced picture emerges. While projective dimension always increases by one, regularity and the a-invariant are preserved in most cases. However, there exists a unique exceptional family: when the path length satisfies n ≡ 1 (mod 3) and C is the specific maximal independent set {x1, x4, …, x_{3k+1}}, both regularity and the a-invariant increase by one (Theorems 5.3, 5.5, and 5.7). This exceptional configuration highlights the sensitivity of homological invariants to the precise local structure of the graph near the suspension vertex.

Methodologically, the proofs blend combinatorial topology and algebra. They heavily utilize Hochster’s formula to translate algebraic questions into topological ones about induced subcomplexes of Δ(G). The Mayer-Vietoris sequence is employed effectively for unions of complexes, particularly in decompositions involving cones over subcomplexes. Discrete Morse theory serves as a powerful tool for gaining precise control over the homology of independence complexes for cycles and paths in the most delicate cases.

In conclusion, this paper demonstrates that selective suspension is a rich and tractable graph operation that induces both rigid, predictable changes and sharp, structure-dependent thresholds in homological invariants. It advances the understanding of how local graph modifications propagate to global algebraic data and provides a refined toolkit for such analyses, suggesting several avenues for future research on other graph classes and operations.