{"paper":{"title":"On the order of regular graphs with fixed second largest eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jack H. Koolen, Jae Young Yang","submitted_at":"2018-09-06T09:03:09Z","abstract_excerpt":"Let $v(k, \\lambda)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $\\lambda$. The Alon-Boppana Theorem implies that $v(k, \\lambda)$ is finite when $k > \\frac{\\lambda^2 + 4}{4}$. In this paper, we show that for fixed $\\lambda \\geq1$, there exists a constant $C(\\lambda)$ such that $2k+2 \\leq v(k, \\lambda) \\leq 2k + C(\\lambda)$ when $k > \\frac{\\lambda^2 + 4}{4}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.01888","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"}