{"paper":{"title":"Expander Evolution Algebras","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Expander evolution algebras connect Cheeger graph expansion to algebraic connectivity and achieve the sharp Alon-Boppana eigenvalue bound over the complex numbers.","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Piero Giacomelli","submitted_at":"2026-05-12T19:22:17Z","abstract_excerpt":"We introduce \\emph{expander evolution algebras} (EEAs), a class of nonassociative algebras defined over an arbitrary field $\\K$ in which the underlying undirected loopless graph of the algebra -- in the sense of Kowalski -- is an expander graph in the classical sense of Cheeger. Starting from the formal graph definition of Kowalski and the algebraic framework of Tian, we establish a dictionary between combinatorial expansion and algebraic structure: the Cheeger constant of the associated graph governs connectivity, the subalgebra lattice, the growth of the evolution sequence, and -- over $\\R$ "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Over C we obtain the sharp Alon-Boppana lower bound for the second eigenvalue of the evolution operator, leading to the definition of Ramanujan evolution algebras as optimal expanders.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The underlying undirected loopless graph of the algebra, in the sense of Kowalski, is an expander graph in the classical sense of Cheeger.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Expander evolution algebras are nonassociative algebras whose graphs are expanders, proven connected and simple with Cheeger constant controlling subalgebra structure and spectral gaps over C.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Expander evolution algebras connect Cheeger graph expansion to algebraic connectivity and achieve the sharp Alon-Boppana eigenvalue bound over the complex numbers.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b45f7e7249ed45133013d708bdf36a32561768d3412b93438feca660932e7e71"},"source":{"id":"2605.12672","kind":"arxiv","version":1},"verdict":{"id":"0cb5ffef-8182-48dd-91de-d8b023d6ff79","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:06:23.788184Z","strongest_claim":"Over C we obtain the sharp Alon-Boppana lower bound for the second eigenvalue of the evolution operator, leading to the definition of Ramanujan evolution algebras as optimal expanders.","one_line_summary":"Expander evolution algebras are nonassociative algebras whose graphs are expanders, proven connected and simple with Cheeger constant controlling subalgebra structure and spectral gaps over C.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The underlying undirected loopless graph of the algebra, in the sense of Kowalski, is an expander graph in the classical sense of Cheeger.","pith_extraction_headline":"Expander evolution algebras connect Cheeger graph expansion to algebraic connectivity and achieve the sharp Alon-Boppana eigenvalue bound over the complex numbers."},"references":{"count":15,"sample":[{"doi":"","year":1985,"title":"N. Alon and V.D. Milman, λ1, isoperimetric inequalities for graphs, and superconcentrators,J. Combin. Theory Ser. B38(1985), no. 1, 73–88","work_id":"32bde0b8-312e-4774-bed7-bc77471ceab7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2008,"title":"J. Bourgain and A. Gamburd, Uniform expansion bounds for Cayley graphs of SL2(Fp),Ann. of Math. (2)167(2008), no. 2, 625–642","work_id":"2d1c7752-8589-4d07-9129-e771d939db21","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"On the connection between evolution algebras, random walks and graphs","work_id":"a6de9097-07cd-4f82-b029-707ba62670a5","ref_index":3,"cited_arxiv_id":"1707.05897","is_internal_anchor":true},{"doi":"","year":2013,"title":"Some properties of evolution algebras","work_id":"6fe14bf0-93df-49bf-adec-41509855afec","ref_index":4,"cited_arxiv_id":"1004.1987","is_internal_anchor":true},{"doi":"","year":2014,"title":"On evolution algebras","work_id":"603991fe-ac42-42fd-a607-db20cd0e9f5e","ref_index":5,"cited_arxiv_id":"1004.1050","is_internal_anchor":true}],"resolved_work":15,"snapshot_sha256":"4deb5cdc60b5c3c71b0a91acc00dbc3e187ffe19089e8b85070a3afb278fd750","internal_anchors":5},"formal_canon":{"evidence_count":2,"snapshot_sha256":"208a3255f553b9557975002a2a005f19c3ea2d186cfc2d6f217896dc1849c918"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}