{"paper":{"title":"On the $A_{\\alpha}$-characteristic polynomial of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shunyi Liu, Xiaogang Liu","submitted_at":"2017-11-09T14:05:01Z","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 $A_{\\alpha}$-characteristic polynomial of $G$ is defined to be $$ \\det(xI_n-A_{\\alpha}(G))=\\sum_jc_{\\alpha j}(G)x^{n-j}, $$ where $\\det(*)$ denotes the determinant of $*$, and $I_n$ is the identity matrix of size $n$. The $A_{\\alpha}$-spectrum of $G$ consists of all roots of the $A_{\\alpha}$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.03868","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"}