Group actions and automorphisms of evolution algebras associated to finite graphs

TL;DR

This paper explores the automorphism structure of evolution algebras linked to finite graphs, revealing how graph symmetries influence algebra automorphisms and extending known results to broader classes.

Contribution

It demonstrates a free action of graph symmetry groups on automorphisms and provides explicit descriptions, extending previous results on perfect evolution algebras.

Findings

Group of graph symmetries acts freely on automorphisms

Explicit description of automorphism set based on graph symmetries

Finite automorphism sets when automorphisms are induced by graph symmetries

Abstract

Given an evolution algebra associated to a connected finite graph $\Gamma$, we exhibit a free action of the group of symmetries of $\Gamma$ on the set of automorphisms of the algebra. This allows us to explicitly describe this set and we prove that a sufficient condition for it to be finite is that every automorphism is induced by a graph symmetry. Consequently, we extend a known result about perfect evolution algebras to other families.