{"paper":{"title":"An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.CO"],"primary_cat":"cs.DM","authors_text":"Luca Trevisan, Nikhil Srivastava","submitted_at":"2017-07-20T03:46:36Z","abstract_excerpt":"We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if $G$ is an $n$-node weighted undirected graph of average combinatorial degree $d$ (that is, $G$ has $dn/2$ edges) and girth $g> 2d^{1/8}+1$, and if $\\lambda_1 \\leq \\lambda_2 \\leq \\cdots \\lambda_n$ are the eigenvalues of the (non-normalized) Laplacian of $G$, then \\[ \\frac {\\lambda_n}{\\lambda_2} \\geq 1 + \\frac 4{\\sqrt d} - O \\left( \\frac 1{d^{\\frac 58} }\\right) \\] (The Alon-Boppana theorem implies that if $G$ is unweighted and $d$-regular, then $\\frac {\\lambda_n}{\\lambda_2} \\geq 1 + \\frac 4"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06364","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"}