{"paper":{"title":"An Improved Upper Bound for the Bilu-Linial Conjecture via Interlacing Families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xinyue Zhang, Zhiqiang Xu","submitted_at":"2026-06-27T08:09:09Z","abstract_excerpt":"The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\\sigma$ such that the spectral radius of the signed adjacency matrix $A_\\sigma$ satisfies $\\rho(A_\\sigma)\\le 2\\sqrt{d-1}$. Bilu and Linial also proved the weaker bound $O(\\sqrt{d\\log^3 d})$ for graphs of maximum degree $d$. Marcus, Spielman, and Srivastava confirmed the conjecture in the case of $d$-regular bipartite graphs. In this paper, we prove that every graph of maximum degree $d$ has a signing $\\sigma$ such that $$\\rho(A_\\sigma)\\le 2\\sqrt{3(d-1)}.$$ This removes the polylogarithmic factor from the estimate"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28797","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28797/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}