Endowing evolution algebras with properties of discrete structures (Q2043312)

In this paper, the authors delve into the basic properties characterizing the directed graph that is uniquely associated to an evolution algebra, and which was introduced in [\textit{J. P. Tian}, Evolution algebras and their applications. Berlin: Springer (2008; Zbl 1136.17001)]. These properties are conveniently translated to algebraic concepts and results concerning the evolution algebra under consideration. In particular, the authors focus on the adjacency of graphs, whose immediate translation into algebraic language enables them to introduce the concepts of adjacency, walk, trail, circuit, path and cycle of an evolution algebra. Also the notions of strongly and weakly connected evolution algebras are introduced as the algebraic equivalences of the same concepts in graph theory. It enables the authors to introduce the notions of distance, girth, circumference, eccentricity, center, radio, diameter and geodesic of an evolution algebra, together with the concepts of Eulerian and Hamiltonian evolution algebras. Some basic results on these topics are then described. The relationship among all of these notions and their analogous in graph theory are visually illustrated throughout the paper.