{"paper":{"title":"Graphs determined by their $A_{\\alpha}$-spectra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiqiu Lin, Jie Xue, Xiaogang Liu","submitted_at":"2017-09-04T02:44:38Z","abstract_excerpt":"Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) $$ for any real $\\alpha\\in [0,1]$. The collection of eigenvalues of $A_{\\alpha}(G)$ together with multiplicities are called the \\emph{$A_{\\alpha}$-spectrum} of $G$. A graph $G$ is said to be \\emph{determined by its $A_{\\alpha}$-spectrum} if all graphs having the same $A_{\\alpha}$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_{\\alpha}$-spectrum for $0\\leq\\alpha<1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00792","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"}