{"paper":{"title":"Polynomial properties on large symmetric association schemes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hiroshi Nozaki","submitted_at":"2013-05-11T20:28:10Z","abstract_excerpt":"In this paper we characterize \"large\" regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that \"large\" association schemes become $P$-polynomial association schemes. Our results are summarized as follows. Let $G=(V,E)$ be a connected $k$-regular graph with $d+1$ distinct eigenvalues $k=\\theta_0>\\theta_1>\\cdots>\\theta_d$. Since the diameter of $G$ is at most $d$, we have the Moore bound \\[ |V| \\leq M(k,d)=1+k \\sum_{i=0}^{d-1}(k-1)^i. \\] Note that if $|V|> M(k,d-1)$ holds, the diameter of $G$ is equal to $d$. L"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.2539","kind":"arxiv","version":3},"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"}