On the isomorphisms between evolution algebras of graphs and random walks

Autor(es)

Cadavid, Paula ^[1] ; Rodino Montoya, Mary Luz ^[2] ; Rodriguez, Pablo M. ^[3]

Afiliação do(s) autor(es)

^[1] Univ Fed ABC, Ctr Matemat Comp & Cognicao, Santo Andre, SP - Brazil

^[2] Univ Antioquia, Inst Matemat, Medellin - Colombia

^[3] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: LINEAR & MULTILINEAR ALGEBRA; v. 69, n. 10 JULY 2019.
• Citações Web of Science: 2

Resumo

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. (AU)

• Processo FAPESP: 16/11648-0 - Teoremas limite e resultados de transição de fase para modelos de propagação de informação em grafos
• Beneficiário: Pablo Martin Rodriguez
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 17/10555-0 - Modelagem estocástica de sistemas interagentes
• Beneficiário: Fabio Prates Machado
• Modalidade de apoio: Auxílio à Pesquisa - Temático