{"paper":{"title":"Connected signed graphs with given inertia indices and given girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Beiyan Liu, Fang Duan","submitted_at":"2025-05-13T13:10:19Z","abstract_excerpt":"Suppose that $\\Gamma=(G, \\sigma)$ is a connected signed graph with at least one cycle. The number of positive, negative and zero eigenvalues of the adjacency matrix of $\\Gamma$ are called positive inertia index, negative inertia index and nullity of $\\Gamma$, which are denoted by $i_+(\\Gamma)$, $i_-(\\Gamma)$ and $\\eta(\\Gamma)$, respectively. Denoted by $g$ the girth, which is the length of the shortest cycle of $\\Gamma$. We study relationships between the girth and the negative inertia index of $\\Gamma$ in this article. We prove $i_{-}(\\Gamma)\\geq \\lceil\\frac{g}{2}\\rceil-1$ and extremal signed"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.08539","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/2505.08539/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"}