{"paper":{"title":"Inertia indices of signed graphs with given cyclomatic number and given number of pendant vertices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.SP","authors_text":"Fang Duan, Jie Pu","submitted_at":"2025-06-29T06:31:42Z","abstract_excerpt":"Let $\\Gamma=(G, \\sigma)$ be a signed graph of order $n$ with underlying graph $G$ and a sign function $\\sigma: E(G)\\rightarrow \\{+, -\\}$. Denoted by $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ the positive inertia index, the cyclomatic number and the number of pendant vertices of $\\Gamma$, respectively. In this article, we prove that $i_+(\\Gamma)$, $\\theta(\\Gamma)$ and $p(\\Gamma)$ are related by the inequality $i_+(\\Gamma)\\geq \\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$. Furthermore, we completely characterize the signed graph $\\Gamma$ for which $i_+(\\Gamma)=\\frac{n-p(\\Gamma)}{2}-\\theta(\\Gamma)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.23112","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/2506.23112/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"}