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

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 $t\ge 2$.