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

Abstract

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.

Recent progress in the research of evolution algebras associated to graphs

“…In this case we suggest to explore the twin partition of the graph. This has been useful to deal with other issues like the derivations space of an evolution algebra associated to a graph, see [2]. A first example in this direction is the family of trees of diameter 3 considered by [3], when we show that the only evolution homomorphism between A(G) and A RW (G) is the null map.…”

Using the Evolution Operator to Classify Evolution Algebras

“…In this paper, we will focus on the study of this operator, specifically in the case where it is a derivation. The space of derivations of an evolution algebra is a frequent subject of study in the literature [14][15][16][17], although obtaining a complete characterization of this space is still an open question. On the other hand, there are also several studies carried out with the aim of classifying certain evolution algebras [18][19][20].…”

The Derivations of Evolution Algebras under a Given Condition

“…Let E be a vector space over a field K with a basis   The basic properties and some classes of evolution algebras were studied as well in [1,8,13]. In [7,9], the space of a derivation of evolution algebra with non-singular matrices and with matrices of rank 1 n  have been described. This work will be described the space of derivations of evolution algebra with matrix of rank 2 n  under a certain possible conditions…”