اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the isomorphisms between evolution algebras of graphs and random walks
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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.
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The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the deriv
ations of this class of algebras for fields of any characteristic.
101 -
Yolanda Cabrera Casado
, Paula Cadavid
, Mary Luz Rodi~no Montoya andn Pablo M. Rodriguez
2019
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 graphs 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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