Analysis shows that graphs with certain primes have a maximum diameter of 6, indicating sharp characteristics within their structure.
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, that this bound is sharp, and that those with diameter 2 are precisely the vertex diameter-2-critical graphs.
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Hien et al. (2026) studied this question.
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