Mathematical tools for the future: Graph Theory and graphicable algebras

Núñez, Juan; Silvero, Marithania; Villar, María Trinidad

doi:10.1016/j.amc.2012.12.004

Cited by 17 publications

(21 citation statements)

References 4 publications

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“…Our results cover a wide range of finite and infinite graphs, including the families of finite graphs that were recently considered by [14]. Indeed, we point out that this work provides a contribution to the theory initiated by [13,14].…”

Section: Introductionmentioning

confidence: 65%

“…More than ten years have passed since the first papers on this topic appeared in Mathematics literature, and a lot of research effort has been devoted to explore the connections between this abstract object and concepts of other fields. We refer the reader to [4,5,6,7,17] for a survey of properties and results of general evolution algebras; to [2,3,8,13,14] for a connection between evolution algebras and graphs; and to [9,11,12,19,16] for a review of results with relevance in genetics and other applications.…”

Section: Introductionmentioning

confidence: 99%

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The connection between evolution algebras, random walks and graphs

2019

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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.

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Section: Introductionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

The connection between evolution algebras, random walks and graphs

2019

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“…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.…”

Section: Resultsmentioning

confidence: 99%

“…, 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.…”

Section: Introductionmentioning

confidence: 99%

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

Cadavid

Montoya

Rodríguez

2018

Linear and Multilinear Algebra

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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.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On the isomorphisms between evolution algebras of graphs and random walks

Cadavid

Montoya

Rodríguez

2019

Linear and Multilinear Algebra

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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. ∞ 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.

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Mathematical tools for the future: Graph Theory and graphicable algebras

Cited by 17 publications

References 4 publications

The connection between evolution algebras, random walks and graphs

The connection between evolution algebras, random walks and graphs

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

On the isomorphisms between evolution algebras of graphs and random walks

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