On the isomorphisms between evolution algebras of graphs and random walks

TL;DR

This paper explores the relationship between two types of evolution algebras derived from graphs, establishing conditions for their isomorphism and isotopy, and analyzing both finite and infinite graph cases with new theoretical insights.

Contribution

It provides new properties relating evolution algebras of graphs, including conditions for isomorphism, a complete description for finite non-singular graphs, and conjectures for singular graphs.

Findings

All graphs' evolution algebras are strongly isotopic.

Conditions for algebra isomorphism depend on graph properties.

Complete classification for finite non-singular graphs.

Abstract

Evolution algebras are non-associative algebras inspired from biological phenomena, with applications to or connections with different mathematical fields. There are two natural ways to define an evolution algebra associated to a given graph. While one takes into account only the adjacencies of the graph, the other includes probabilities related to the symmetric random walk on the same graph. In this work we state new properties related to the relation between these algebras, which is one of the open problems in the interplay between evolution algebras and graphs. On the one hand, we show that for any graph both algebras are strongly isotopic. On the other hand, we provide conditions under which these algebras are or are not isomorphic. For the case of finite non-singular graphs we provide a complete description of the problem, while for the case of finite singular graphs we state a conjecture supported by examples and partial results. The case of graphs with an infinite number of vertices is also discussed. As a sideline of our work, we revisit a result existing in the literature about the identification of the automorphism group of an evolution algebra, and we give an improved version of it.