{"paper":{"title":"A note on the positive semidefinitness of $A_\\alpha (G)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo, Vladimir Nikiforov","submitted_at":"2016-11-06T18:27:26Z","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] $, write $A_{\\alpha}\\left( G\\right) $ for the matrix \\[ A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) . \\] Let $\\alpha_{0}\\left( G\\right) $ be the smallest $\\alpha$ for which $A_{\\alpha}(G)$ is positive semidefinite. It is known that $\\alpha_{0}\\left( G\\right) \\leq1/2$. The main results of this paper are:\n  (1) if $G$ is $d$-regular then \\[ \\alpha_{0}=\\frac{-\\lambda_{\\min}(A(G))}{d-\\lambda_{\\min}(A(G))}, \\] wher"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.01818","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"}