{"paper":{"title":"Complex unit gain bicyclic graphs with rank 2, 3 or 4","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ligong Wang, Peng Xiao, Yong Lu","submitted_at":"2015-11-24T06:46:08Z","abstract_excerpt":"A $\\mathbb{T}$-gain graph is a triple $\\Phi=(G,\\mathbb{T},\\varphi)$ consisting of a graph $G=(V,E)$, the circle group $\\mathbb{T}=\\{z\\in C: |z|=1\\}$ and a gain function $\\varphi:\\overrightarrow{E}\\rightarrow \\mathbb{T}$ such that $\\varphi(e_{ij})=\\varphi(e_{ji})^{-1}=\\overline{\\varphi(e_{ji})}$. The rank of $\\mathbb{T}$-gain graph $\\Phi$, denoted by $r(\\Phi)$, is the rank of the adjacency matrix of $\\Phi$. In 2015, Yu, Qu and Tu [ G. H. Yu, H. Qu, J. H. Tu, Inertia of complex unit gain graphs, Appl. Math. Comput. 265(2015) 619--629 ] obtained some properties of inertia of a $\\mathbb{T}$-gain g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07589","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"}