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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1907.07837","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-18T01:54:56Z","cross_cats_sorted":[],"title_canon_sha256":"20cac42c9890725d5c6199b7e6de121f9b57b0f5044f835b1cdb0e6b0ae3371b","abstract_canon_sha256":"6eb8361f6c492f12c3200c7bfcd98fd1e92301123bc2f9edb9150252714119c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:16.042673Z","signature_b64":"AvHbJhB5n5uSq4EAssyTHdjMLqVHkS7K0yRRqWHYtagpjU6jZUEMOVgV49kLOm3sZYXfz24t1tgVlxwQbmhLCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5ba3785766555b7b7361d181209dcb21345240496a0008deb562afed380d9750","last_reissued_at":"2026-05-17T23:40:16.042047Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:16.042047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $2n-2c(G) \\leq r(G, \\sigma)+2\\alpha(G) \\leq 2n$. 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