On homological invariants and Cohen-Macaulayness of closed neighborhood ideals

TL;DR

This paper investigates algebraic invariants of closed neighborhood ideals of graphs, establishing formulas for regularity in chordal graphs, exploring inequalities in bipartite and well-covered graphs, and characterizing Cohen-Macaulay cases.

Contribution

It provides new formulas for regularity in chordal graphs, explores inequalities in bipartite and well-covered graphs, and characterizes Cohen-Macaulayness of the associated rings.

Findings

Regularity equals vertex cover number in chordal graphs.

Inequality reg(S/NI(G)) ≥ τ(G) holds for bipartite and well-covered graphs.

Characterization of Cohen-Macaulay closed neighborhood ideals.

Abstract

Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal. For any chordal graph $G$, we show that $\text{reg}(S/NI(G))=\tau(G)$, where $\tau(G)$ denotes the vertex cover number of $G$. This generalizes the corresponding result for trees shown in [3], as in trees $\tau(G)$ is the same as the matching number of $G$. When $G$ is a bipartite graph or a very well-covered graph, we notice that $\text{reg}(S/NI(G))\geq \tau(G)$ and that this inequality can be strict in general. Moreover, we describe the projective dimension of $S/NI(G)$ for some families of graphs. Finally, we give a characterization of very well-covered graphs $G$ for which the ring $S/NI(G)$ is Cohen-Macaulay.