Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

TL;DR

This work connects graph matching theory to the algebraic invariants of the closed neighborhood ideal $NI(G)$. It proves, for forests, that the Castelnuovo-Mumford regularity satisfies $\mathrm{reg}(R/NI(G))=a_G$, confirming a conjecture by Sharifan and Moradi, and establishes the universal lower bound $\mathrm{reg}(R/NI(G))\ge a_G$ for all graphs via topological methods. It further shows that if $G$ contains a simplicial vertex, $NI(G)$ admits a Betti splitting, yielding a lower bound $\mathrm{pd}(R/NI(G))\ge a_G$ in forests and unicyclic graphs, while the whisker construction yields explicit equalities $\mathrm{reg}(R'/NI(G'))=\mathrm{pd}(R'/NI(G'))=a_{G'}$. Together, these results illuminate how the combinatorial structure of $G$ governs the homological invariants of $NI(G)$, with sharp bounds and notable exceptions (e.g., certain chordal graphs) that motivate further study.

Abstract

Let $G$ be a finite simple graph and let $NI(G)$ denote the closed neighborhood ideal of $G$ in a polynomial ring $R$. We show that if $G$ is a forest, then the Castelnuovo-Mumford regularity of $R/NI(G)$ is the same as the matching number of $G$, thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of $G$ provides a lower bound for the Castelnuovo-Mumford regularity of $R/NI(G)$ for any $G$. Furthermore, we prove that, if $G$ contains a simplicial vertex, then $NI(G)$ admits a Betti splitting, and consequently, we show that the projective dimension of $R/NI(G)$ is also bounded below by the matching number of $G$, if $G$ is a forest or a unicyclic graph.

Figures (1)

• Figure 1: A rooted tree $T$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• Theorem 2.5: Fröberg's theorem
• Corollary 2.6