{"paper":{"title":"A note on the $A_{\\alpha}$-spectral radius of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiqiu Lin, Jie Xue, Xing Huang","submitted_at":"2018-05-15T14:31:47Z","abstract_excerpt":"Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\\alpha\\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_{\\alpha}(G)$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\\leq i\\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05808","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"}