The connection between evolution algebras, random walks and graphs

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. ∞ k=1 c ik = 1,for any i, k, then A is called Markov evolution algebra. The name is due to that there is an interesting one-to-one correspondence between A and a discrete time Markov chain (X n ) n≥0 with 2010 Mathematics Subject Classification. 05C25, 17D92, 17D99, 05C81.

Characterization theorems for the spaces of derivations of evolution algebras associated to graphs

“…The first definitions of evolution algebras associated to some families of graphs have been formalized by [17,18], were the authors also study properties related to these algebras. More recently, [3,4] study the existence of isomorphisms between these algebras and the evolution algebras associated to the random walk on the same graph. This is a subject that has been remained as an open question since 2008, see [20,21] for further details.…”

“…, n}. On the other hand, we continue in studying evolution algebras associated to graphs, a subject which was initiated by [18,20], and considered more recently by [3,4]. We shall see that the twin partition of the given graph is crucial for characterizing the space of derivations of its respective evolution algebra.…”

On the isomorphisms between evolution algebras of graphs and random walks

“…The best general reference of the subject is [14], where the reader can found a review of preliminary definitions and properties, connections with other fields of mathematics, and a list of interesting open problems some of which remain unsolved so far. We refer the reader also to [15] for an update of open problems in the Theory of Evolution Algebras, and to [2]- [9] and references therein for an overview of recent results on this topic. Formally, an evolution algebra is defined as follows.…”

“…evolution algebras and graphs, has attained the attention of many researchers in recent years. For a review of recent results, see for instance [2,3,6,7,11,12], and references therein. The rest of the section is subdivided into two parts.…”

“…The aim of this paper is to contribute with the discussion about the relation between the algebras A RW (G) and A(G) for a given graph G. We emphasize that this is one of the open problems stated by [14,15], and which has been addressed partially by [2]. Our approach will be the statement of conditions under which we can guarantee the existence or not of isomorphisms between these evolution algebras.…”