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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.06364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-20T03:46:36Z","cross_cats_sorted":["cs.DS","math.CO"],"title_canon_sha256":"ee8d37a38b78a979be9141f6cf5a3d455ac6b670006fd11078e586a6f482ab56","abstract_canon_sha256":"344fadc7f203212a9ffb3337cc0f48623c1a8199a2137b63dae2ba9ad9cfa5f1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:54.011242Z","signature_b64":"wn6bEh4K1BQeS69+aYL/kTjiOBNcB13IWSNB70oaC+RD4cZ4MisDjigcn83HArl0Eie5mscnn4R1M8uUPgtBBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01818884a0338376d471bdb5ed10d12528dd325c51f3ab96a07578a8111836c2","last_reissued_at":"2026-05-18T00:39:54.010381Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:54.010381Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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} - 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