{"paper":{"title":"Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan Rooney, Michael Tait, Shahla Nasserasr, Shaun Fallat, Veronika Furst, Wayne Barrett","submitted_at":"2024-11-19T23:05:43Z","abstract_excerpt":"The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\\overline{G}) = |E(\\overline{G})| \\leq \\lfloor n/2 \\rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\\overline{G}) \\leq n-3$ satisfies $q(G) = 2$. We show that this conjecture is true if $\\overline{G}$ is bipartite and in other sporadic cases. Furthermore, we characterize $G$ with $\\overline{G}$ bipartite and $e(\\overline{G}) = n-2$ for which $q(G) > 2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.12917","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.12917/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}