If the maximal homogeneous ideal is an associated prime of the tth power of the closed neighborhood ideal of G, then the diameter of G is at most 7t-8, and this bound is sharp.
Brodmann, Asymptotic stability of (M/I^nM) , Proc
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Associated primes of powers of closed neighborhood ideals and diameters of graphs
If the maximal homogeneous ideal is an associated prime of the tth power of the closed neighborhood ideal of G, then the diameter of G is at most 7t-8, and this bound is sharp.