Expander Evolution Algebras

TL;DR

This paper introduces expander evolution algebras (EEAs), linking combinatorial graph expansion properties with algebraic structures, spectral gaps, and connectivity, and explores their properties over various fields.

Contribution

It establishes a novel connection between expander graph theory and evolution algebras, including spectral bounds and construction methods from Cayley graphs.

Findings

EEAs are always connected and simple over any field.

Over e9 and c9, EEAs exhibit spectral properties related to Ramanujan graphs.

Constructs families of EEAs from Cayley graphs of finite groups.

Abstract

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.