{"paper":{"title":"On the $A_{\\alpha}$-spectra of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Germain Past\\'en, Oscar Rojo, Ricardo L. Soto, Vladimir Nikiforov","submitted_at":"2016-09-03T15:14:49Z","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 every real $\\alpha\\in\\left[ 0,1\\right],$ define the matrix $A_{\\alpha}\\left(G\\right) $ as \\[ A_{\\alpha}\\left(G\\right) =\\alpha D\\left(G\\right) +(1-\\alpha)A\\left(G\\right) \\] where $0\\leq\\alpha\\leq1$.\n  This paper gives several results about the $A_{\\alpha}$-matrices of trees. In particular, it is shown that if $T_{\\Delta}$ is a tree of maximal degree $\\Delta,$ then the spectral radius of $A_{\\alpha}(T_{\\Delta})$ satisfies the tight inequality \\[ \\rho(A_{\\alpha}(T_{\\Delta}))<\\alpha\\Del"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.00835","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"}