{"paper":{"title":"Equitable partitions for Ramanajun graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mohsen Alinejad, Sanaz Fulad","submitted_at":"2021-07-24T09:18:05Z","abstract_excerpt":"For d-regular graph G, an edge-signing sigma:E(G) \\rightarrow {-1,1} is called a good signing if the absolute eigenvalues of adjacency matrix are at most 2 \\sqrt{d-1}. Bilu-Linial conjectured that for each regular graph there exists a good signing. In this paper, by using new concept \"Equitable Partition\", we solve the Bilu-Linial Conjecture for some cases. We show that how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G, then we can find a good signing for a 2-lift of G."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.11563","kind":"arxiv","version":2},"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/2107.11563/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"}