Associated primes of powers of closed neighborhood ideals and diameters of graphs

TL;DR

This paper establishes a precise relationship between the associated primes of powers of closed neighborhood ideals of graphs and the graph's diameter, providing sharp bounds for all relevant powers.

Contribution

It proves that if the maximal homogeneous ideal is an associated prime of the t-th power of the closed neighborhood ideal, then the graph's diameter is bounded by 7t-8, and this bound is sharp.

Findings

Bound on graph diameter based on associated primes

Sharpness of the diameter bound for all t ≥ 2

Connection between algebraic properties and graph diameter

Abstract

Let $G$ be a simple connected graph and $t \ge 2$ an integer. We prove that if the maximal homogeneous ideal is an associated prime of the $t$th power of the closed neighborhood ideal of $G$, then the diameter of $G$ is at most $7t - 8$. We further show that this bound is sharp for all $t \ge 2$.