On the closed neighborhood ideal of the square of the path graph

TL;DR

This paper investigates the algebraic properties of the closed neighborhood ideal of the square of a path graph, computing key invariants and characterizing Cohen-Macaulay conditions.

Contribution

It provides explicit calculations of algebraic invariants and characterizations for Cohen-Macaulayness of these ideals, advancing understanding in algebraic graph theory.

Findings

Computed height, projective dimension, and regularity of the ideals.

Proved the ideals are sequentially Cohen-Macaulay.

Characterized when the ideals are Cohen-Macaulay.

Abstract

We consider the closed neighborhood ideal of square of the path graph and study some of its algebraic and homological invariants. We compute the height, the projective dimension and the Castelnuovo-Mumford regularity. We prove that these ideals are sequentially Cohen-Macaulay and characterize when they are Cohen-Macaulay.