Algebraic Invariants of Edge Ideals Under Suspension

TL;DR

This paper investigates how algebraic invariants of edge ideals change under graph suspension operations, revealing that certain invariants are preserved or systematically altered depending on the suspension type and the graph's structure.

Contribution

It introduces selective suspension constructions over minimal vertex covers and maximal independent sets, providing detailed invariance behavior and explicit formulas for these algebraic invariants.

Findings

Regularity is preserved under cover suspension while projective dimension increases by one.

In suspension over maximal independent sets, projective dimension increases by one, but regularity is often preserved.

Paths and cycles exhibit specific invariant changes, with paths showing simultaneous increases in invariants.

Abstract

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.