{"paper":{"title":"$A_{\\alpha}$-spectrum of a graph obtained by copies of a rooted graph and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo","submitted_at":"2017-04-22T01:07:01Z","abstract_excerpt":"Given a connected graph $R$ on $r$ vertices and a rooted graph $H,$ let $R\\{H\\}$ be the graph obtained from $r$ copies of $H$ and the graph $R$ by identifying the root of the $i-th$ copy of $H$ with the $i-th$ vertex of $R$. Let $0\\leq\\alpha\\leq1,$ and let \\[ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) \\] where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. A basic result on the $A_{\\alpha}-$ spectrum of $R\\{H\\}$ is obtained. This result is used to prove that if $H=B_{k}$ is a generalized Bethe tree on $k$ levels, then the eig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06730","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"}