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Denote by $\\theta(G)=|E(G)|-|V(G)|+\\omega(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $\\omega(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(\\Phi)$ in terms of $r(G)$, that is, $r(G)-2\\theta(G)\\leq r(\\Phi)\\leq r(G)+2\\theta"},"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":"1711.11448","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-11-29T14:07:14Z","cross_cats_sorted":[],"title_canon_sha256":"231ae28599c3c89ea8f9fe83aecc841b223d3d420b4aa59e7f837d16b78b02e6","abstract_canon_sha256":"30a8e34a602107e5816cd6a48aadb557ed394881b650ee2336cd43e8e6ea5cd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:11.565613Z","signature_b64":"DNpE9jxlEpIQD0V38CW9+EwjtpoWUlLthq/7+kUXyumIoCWRnjR+pZNaYoNT8gFpZe355RTqYzdV+opgZUiACA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9bcf6933f754ba8351414f34f56e586e7ea394a4a3eef0cb7852f66da13dd4b","last_reissued_at":"2026-05-18T00:29:11.565082Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:11.565082Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The rank of a complex unit gain graph in terms of the rank of its underlying graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ligong Wang, Qiannan Zhou, Yong Lu","submitted_at":"2017-11-29T14:07:14Z","abstract_excerpt":"Let $\\Phi=(G, \\varphi)$ be a complex unit gain graph (or $\\mathbb{T}$-gain graph) and $A(\\Phi)$ be its adjacency matrix, where $G$ is called the underlying graph of $\\Phi$. 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