{"paper":{"title":"Relation between the H-rank of a mixed graph and the rank of its underlying graph","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chen Chen, Minjie Zhang, Shuchao Li","submitted_at":"2018-12-13T08:24:44Z","abstract_excerpt":"Given a simple graph $G=(V_G, E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\\widetilde{G}$ is obtained from $G$ by orienting some of its edges. Let $H(\\widetilde{G})$ denote the Hermitian adjacency matrix of $\\widetilde{G}$ and $A(G)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\\widetilde{G}$ (resp. $G$), written as $rk(\\widetilde{G})$ (resp. $r(G)$), is the rank of $H(\\widetilde{G})$ (resp. $A(G)$). Denote by $d(G)$ the dimension of cycle spaces of $G$, that is $d(G) = |E_G|-|V_G|+\\omega(G)$, where $\\omega(G),$ denotes the number of connected components of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.05309","kind":"arxiv","version":2},"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"}