{"paper":{"title":"On the permanental nullity and matching number of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong-Jian Lai, Tingzeng Wu","submitted_at":"2016-03-10T00:17:46Z","abstract_excerpt":"For a graph $G$ with $n$ vertices, let $\\nu(G)$ and $A(G)$ denote the matching number and adjacency matrix of $G$, respectively. The permanental polynomial of $G$ is defined as $\\pi(G,x)={\\rm per}(Ix-A(G))$. The permanental nullity of $G$, denoted by $\\eta_{per}(G)$, is the multiplicity of the zero root of $\\pi(G,x)$. In this paper, we use the Gallai-Edmonds structure theorem to derive a concise formula which reveals the relationship between the permanental nullity and the matching number of a graph. Furthermore, we prove a necessary and sufficient condition for a graph $G$ to have $\\eta_{per}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.03109","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"}