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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2605.12672 |
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| _version_ | 1866918498362458112 |
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| author | Giacomelli, Piero |
| author_facet | Giacomelli, Piero |
| contents | 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 $\R$ and $\C$ -- the spectral gap of the evolution operator. Over a general field $\K$ we prove that EEAs are always connected and simple (as evolution algebras), carry no proper large evolution subalgebras, and that every generator of a \emph{symmetric} EEA is algebraically persistent. Over $\C$ we obtain the sharp Alon--Boppana lower bound for the second eigenvalue of the evolution operator, leading to the definition of \emph{Ramanujan evolution algebras} as optimal expanders. We also construct families of EEAs from Cayley graphs of finite groups. We close with open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12672 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Expander Evolution Algebras Giacomelli, Piero Rings and Algebras Combinatorics 17D92, 05C50, 15A18, 60J10 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 $\R$ and $\C$ -- the spectral gap of the evolution operator. Over a general field $\K$ we prove that EEAs are always connected and simple (as evolution algebras), carry no proper large evolution subalgebras, and that every generator of a \emph{symmetric} EEA is algebraically persistent. Over $\C$ we obtain the sharp Alon--Boppana lower bound for the second eigenvalue of the evolution operator, leading to the definition of \emph{Ramanujan evolution algebras} as optimal expanders. We also construct families of EEAs from Cayley graphs of finite groups. We close with open problems. |
| title | Expander Evolution Algebras |
| topic | Rings and Algebras Combinatorics 17D92, 05C50, 15A18, 60J10 |
| url | https://arxiv.org/abs/2605.12672 |