
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 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 and -- the spectral gap of the evolution operator. Over a general field 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 we…
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