{"paper":{"title":"Regularity of bicyclic Graphs and their powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Beatrice Picone, Navid Nemati, Sepehr Jafari, Yairon Cid-Ruiz","submitted_at":"2018-02-20T17:04:32Z","abstract_excerpt":"Let $I(G)$ be the edge ideal of a bicyclic graph. In this paper, we characterize the Castelnuovo-Mumford regularity of $I(G)$ in terms of the induced matching number of $G$. For the base case of this family of graphs, i.e. dumbbell graph, we explicitly compute the induced matching number. Moreover, we prove that $ \\operatorname{reg}(I(G)^q)=2q+\\operatorname{reg}(I(G))-2$, for all $ q\\geq 1 $, when $ G $ is a dumbbell graph with a connecting path having no more than two vertices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.07202","kind":"arxiv","version":4},"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"}