{"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 $\\Phi$. The rank of $\\Phi$, denoted by $r(\\Phi)$, is the rank of $A(\\Phi)$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.11448","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"}