{"paper":{"title":"Diameter and Girth of Zero Divisor Graph of Multiplicative Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Sachin Sarode, Vinayak Joshi","submitted_at":"2013-10-17T10:50:05Z","abstract_excerpt":"In this paper, we study the zero divisor graph $\\Gamma^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, $\\Gamma^m(L)$ contains a cycle and $gr(\\Gamma^m(L)) = 3$. This essentially proves that for a reduced ring R with more than two minimal primes, $gr(\\mathbb{AG}(R))) = 3$ which settles the conjecture of Behboodi and Rakeei [9]. Further, we have characterized the diameter of $\\Gamma^m(L)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.4653","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}