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.

It is well-known that the space of derivations of n-dimensional evolution algebras with non-singular matrices is zero. On the other hand, the space of derivations of evolution algebras with matrices of rank n − 1 has also been completely described in the literature. In this work we provide a complete description of the space of derivations of evolution algebras associated to graphs, depending on the twin partition of the graph. For graphs without twin classes with at least three elements we prove that the space of derivations of the associated evolution algebra is zero. Moreover, we describe the spaces of derivations for evolution algebras associated to the remaining families of finite graphs. It is worth pointing out that our analysis includes examples of finite dimensional evolution algebras with matrices of any rank.2010 Mathematics Subject Classification. 17A36, 05C25, 17D92, 17D99.

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. ∞ k=1 c ik = 1, for any i, then A is called a Markov evolution algebra. In this case, there is an interesting correspondence between the algebra A and a discrete time Markov chain (X n ) n≥0 with states space {x 1 , x 2 , . . . , x n , . . .} and transition probabilities given by: c ik := P(X n+1 = x k |X n = x i ), 2010 Mathematics Subject Classification. 05C25, 17D92, 17D99, 05C81.

We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph we prove that the space of derivations is zero. For the remaining families of evolution algebras we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graph's structure impacts in the characterization of derivations for a given evolution algebra. Therefore our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible 3-dimensional evolution algebras.

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