Associated primes of the second power of closed neighborhood ideals of graphs

TL;DR

This paper investigates the properties of graphs related to the associated primes of the second power of their closed neighborhood ideals, revealing diameter constraints and criticality conditions.

Contribution

It characterizes graphs with the maximal homogeneous ideal as an associated prime of the second power of their closed neighborhood ideals, including diameter bounds and criticality conditions.

Findings

Graphs with the maximal homogeneous ideal as an associated prime have diameter at most 6.

Graphs with diameter 2 are vertex diameter-2-critical.

Such properties help understand the algebraic and combinatorial structure of these graphs.

Abstract

We study simple graphs for which the maximal homogeneous ideal is an associated prime of the second power of their closed neighborhood ideals. In particular, we show that such graphs must have diameter at most $6$, and that those with diameter $2$ must be vertex diameter-$2$-critical.