On the connection between evolution algebras, random walks and graphs

TL;DR

This paper investigates the relationship between evolution algebras, random walks, and graphs, revealing how algebraic structures can model probabilistic processes on graphs and potentially advance the study of Markov evolution algebras.

Contribution

It establishes new connections between evolution algebras and random walks on graphs, expanding the algebraic framework for analyzing Markov processes.

Findings

Relation between evolution algebras and random walks on graphs

Characterization of evolution algebras induced by graphs

Potential applications to Markov evolution algebras

Abstract

Evolution algebras are a new type of non-associative algebras which are inspired from biological phenomena. A special class of such algebras, called Markov evolution algebras, is strongly related to the theory of discrete time Markov chains. The winning of this relation is that many results coming from Probability Theory may be stated in the context of Abstract Algebra. In this paper we explore the connection between evolution algebras, random walks and graphs. More precisely, we study the relationships between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Given that any Markov chain may be seen as a random walk on a graph we believe that our results may add a new landscape in the study of Markov evolution algebras.