Certain particular families of graphicable algebras

Núñez, Juan; Rodríguez-Arévalo, María Luisa; Villar, María Trinidad

doi:10.1016/j.amc.2014.08.007

Cited by 13 publications

(22 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

See 1 more Smart Citation

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

Cadavid

Montoya

Rodríguez

2018

Linear and Multilinear Algebra

View full text Add to dashboard Cite

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.

show abstract

Section: Resultsmentioning

confidence: 99%

“…These algebras are non-associative algebras and form a special class of genetic algebras. We refer the reader to [4,5,10,17,6] and references therein for an overview of recent results. An n-dimensional evolution algebra is defined as follows.…”

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

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…The Tietze graph [5] has 12 vertices and 18 edges, which likes the Petersen graph. The Tietze graph is maximally non-hamiltonian and it has no hamiltonian cycle, but any two nonadjacent vertices can be connected by a Hamiltonian path.…”

Section: Preliminariesmentioning

confidence: 99%