Expander Evolution Algebras

@misc{giacomelli2026expander,
  title = {{Expander Evolution Algebras}},
  author = {Piero Giacomelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.12672},
  eprint = {2605.12672},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.12672},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Expander Evolution Algebras
AU  - Piero Giacomelli
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2605.12672
UR  - https://arxiv.org/abs/2605.12672
AB  - 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.
ER  -