{"paper":{"title":"The relation between the independence number and rank of a signed graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rong-Xia Hao, Shengjie He","submitted_at":"2019-07-18T01:54:56Z","abstract_excerpt":"A signed graph $(G, \\sigma)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $(G, \\sigma)$. Let $c(G)$, $\\alpha(G)$ and $r(G, \\sigma)$ be the cyclomatic number, the independence number and the rank of the adjacency matrix of $(G, \\sigma)$, respectively. In this paper, we study the relation among the independence number, the rank and the cyclomatic number of a signed graph $(G, \\sigma)$ with order $n$, and prove that $2n-2c(G) \\leq r(G, \\sigma)+2\\alpha(G) \\leq 2n$. Furthermore, the signed graphs that reaching the lower bound are investigated."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07837","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"}