New advances on (pseudo)digraphs and evolution algebras

Computational and Applied Mathematics (2022) 41:148 https://doi.org/10.1007/s40314-022-01858-7 New advances on (pseudo)digraphs and evolution algebras M. Ceballos1 Received: 16 November 2021 / Revised: 18 March 2022 / Accepted: 28 March 2022 © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2022 Abstract In this paper, new advances concerning the link between evolution algebras and (pseudo)digraph are shown. Some important elements that can be read from the (pseudo)digraph that is associated with an evolution algebra are studied. Moreover, several results concerning solvability, nilpotency, and the preservation of them under the graph union operation are proved. To complement the theoretical study, an algorithmic method has been implemented. This is devoted to computing the nilpotency index of a nilpotent evolution algebra using its associated digraph. Keywords (Pseudo)Digraph ·Evolution algebra ·Derived algebra ·Algorithm Mathematics Subject Classification 17D92 ·05C25 ·05C20 ·05C85 ·05C90 ·68W30 · 68R10 1 Introduction Nowadays, one of the most important and relevant research in Mathematics is finding and studying new links between different fields. Alternative techniques and procedures allow researchers to solve many open problems, improve well-known theories, and achieve new results. This paper deals with the link between evolution algebras and Graph Theory. In 2006, Tian and Vojtechovsky introduced evolution algebras (see Tian and Vojtechovsky 2006). Later, Tian set the basic concepts of evolution algebras in Tian (2008). These algebras lie between dynamical systems and non-associative algebras. From an algebraic point of view, evolution algebras are Banach non-associative algebras, and dynamically, they are a discrete dynamical system. In Cabrera et al. (2016), some general algebraic properties of evolution algebras were studied. The authors analyzed evolution ideals, evolution subalgebras, non-degeneracy, and simple and irreducible evolution algebras. There exist many connections between these algebras and other mathematical fields such as graph theory, Communicated by Carlos Hoppen. BM. Ceballos [email protected] 1Departamento de Ingeniería, Universidad Loyola Andalucía, Av. de las Universidades, s/n, 41704 Dos Hermanas, Seville, Spain 0123456789().: V,-vol 123 148 Page 2 of 17 M. Ceballos group theory, stochastic processes, physics, etc. (Rozikov and Tian (2011) can be seen, for example). Moreover, there are many algebraic open problems about these algebras such as their classification. Evolution algebras have only been classified up to dimension 3 (Casas et al. 2014; Cabrera et al. 2017). There is also a classification of some families of nilpotent evolution algebras with dimension less than six (Hegazi and Abdelwahab 2015). The formulation of Mendel’s law is another important application of evolution algebras. In fact, Tian showed in Tian (2008) the close connection between evolution algebras, Markov chains, and non-Mendelian genetics. More concretely, evolution algebras can be applied to the inheritance of organelle genes, for example, to estimate all possible mechanisms to set the homoplasmy of cell populations. In this paper, we will deal with some algebraic notions such as solvability and nilpotency of evolution algebras. Those concepts can be interpreted biologically as the fact that some of the original generators become extinct after a certain number of generations. Currently, Graph Theory is an essential tool to solve a huge number of problems in different mathematical research fields. In this way, semisimple algebras can be studied using graphs, since trees allow to determine the Dynkin diagrams that are associated to such algebras (Serre 1996). Moreover, Graph Theory can be also applied to represent finite-dimensional algebras (Primc 2000). There are several papers in the literature concerning the relation between Graph Theory and evolution algebras. First, in 2008, Tian (2008) described a way to associate graphs and digraphs with evolution algebras. He also dealt with the converse problem. Tian remarked in Tian (2008) that the “intrinsic relation of evolution algebras with graph theory allows to analyze graphs algebraically [...] and graph theory can be used to deal with non-associative algebras”. In 2011, Rozikov and Tian (2011) defined evolution algebras associated with function spaces given by finite and connected graphs. In 2015, Elduque and Labra (2015) used digraphs to study if a finite-dimensional evolution algebra is nil. In 2016, Cabrera et al. (2016) described a digraph related to a non-degenerate evolution algebra, so that the latter is irreducible if and only if the former is connected. More recently, in 2019, Cadavid et al. (2019) described the space of derivations of evolution algebras associated with graphs, depending on a partition of their sets of vertices. One year later (see Cadavid et al. 2020), they dealt with the relationship between the evolution algebra induced by a random walk on a graph and the evolution algebra determined by the same graph. Also, in 2020, Ceballos et al. (2020) described new (pseudo)digraphs associated with evolution algebras, using this association to classify low-dimensional evolution algebras. The main goal of this current paper is to continue with the research line started in Carriazo et al. (2004); Ceballos et al. (2011,2018), where a link among combinatorial structures and Lie or Leibniz algebras was established and, hence, several properties on those nonassociative algebras can be translated into the Graph Theory field and vice versa. This paper extends these studies to the case of evolution algebras to continue with the research line started in Ceballos et al. (2020); Elduque and Labra (2015) analyzing the link among weighted (pseudo)digraphs and evolution algebras. In Ceballos et al. (2020), the authors continued the work started in Carriazo et al. (2004) and developed in Ceballos et al. (2011,2015,2018)and Cáceres et al. (2012), but this time for evolution algebras. The structure of this paper is divided into several sections as follows: Sect. 2reviews some concepts and notation on both, evolution algebras and Graph Theory. Next, Sect. 3recalls the procedure devoted to associating a weighted (pseudo)digraph with an evolution algebra like the one developed in Ceballos et al. (2020) and Elduque and Labra (2015). Next, in Sect. 4, several new results concerning (pseudo)digraphs and evolution algebras are obtained. First, some properties that can be read from the associated (pseudo)digraph are analyzed. Next, we study how the solvability and nilpotency can be translated into the properties of the associated (pseudo)digraph and 123 New advances on (pseudo)digraphs and evolution algebras Page 3 of 17 148 the union operation. After that, Sect. 5deals with the implementation of an algorithm to compute the nilpotency index of a given nilpotent evolution algebra using its associated digraph. Finally, a complexity and computational study is carried out in Sect. 6giving also the computing time and number of operations of each step of the algorithm. 2 Preliminaries This section recalls some preliminary concepts, results, and notations about evolution algebras and graphs. Concerning the former, the reader can consult (Tian 2008). Regarding the latter, Harary (1969) is an introductory reference to Graph Theory. 2.1 Evolution algebras Let Kbe a field and E≡(E,+,·)an algebra over K. We say that Eis an n-dimensional evolution algebra if we can find a basis B={ei}n i=1of the evolution algebra, such that 1) e2 i=n k=1ci,kek,∀1≤i≤n;and 2) ei·ej=0, ∀1≤i= j≤n. The basis Bis called natural basis of the evolution algebra E,ci,jare known as the structure constants and A=(ci,j)is called the structure matrix. From now on, we will study ndimensional evolution algebras Eover the field of complex numbers with natural basis B. Evolution algebras are commutative and flexible (Tian and Vojtechovsky 2006), but they are not associative or power-associative. One can find two different types of trivial evolution algebras: the ones called zero evolution algebras, which are those verifying ei·ej=0, ∀1≤i,j≤nand the ones called non-zero evolution algebras,wheree2 i=ci,iei,∀1≤i≤n. An evolution algebra is non-degenerate if e2 i= 0, ∀1≤i≤n. Otherwise, they are called degenerate. Given an evolution algebra Eand a fixed natural basis B,anelemente∈Eis said to be absolute nilpotent with respect to Bif e2=0, while e∈Eis idempotent if e2=e. The set of all the absolute nilpotent elements of Ewill be denoted by An(E). The annihilator of an evolution algebra Eis defined by Ann(E)={X∈E|X·Y= 0,∀Y∈E}. Clearly, Ann(E)⊂An(E). The derived series of a given finite-dimensional evolution algebra Eis E(1)=E,E(2)=E·E, ..., E(k)=E(k−1)·E(k−1), ... We say that Eis solvable if ∃m∈N,m>1, such that E(m)={0}.IfE(m−1)={0}also holds, then mis known as the solvability index or solvindex of Eand it is said that Eis (m−1)-step solvable. The central series of a given finite-dimensional evolution algebra Eis E<1>=E,E<2>=E·E, ..., E<k>=E<k−1>·E, ... We say that Eis nilpotent if ∃m∈N,m>1, such that E<m>={0}.IfE<m−1>={0} also holds, then mis known as the nilpotency index or nilindex of Eand it is said that Eis (m−1)-step nilpotent. Notice that, for a general algebra, the previous definition corresponds to right nilpotency, but in evolution algebras, both classes nilpotent and right nilpotent are the same. 123 148 Page 4 of 17 M. Ceballos Obviously, every nilpotent evolution algebra is solvable, since E(i)⊆E<i>,∀i∈N. Nilpotent evolution algebras having maximal nilpotency index (n-step) are called filiform. The derived algebra of an evolution algebra Ewill be denoted by DE=E(2)=E<2>. An evolution algebra Eis perfect if Eand DEare equal. 2.2 Graph theory Agraph is a pair G=(V,E),whereVis called the non-empty vertex-set (or node-set) and Eis called edge-set, which is given by unordered pairs of a couple of nodes. It is possible to associate a weight to each edge. In case that Eis given by ordered pairs of vertices, we will say that Gis a directed graph or a digraph. An edge of a digraph connecting a vertex with itself is usually known as a loop. A digraph containing loops will be called pseudodigraph. Throughout the paper, weighted (pseudo)digraphs with possible double edges will be considered. Avertexv∈Vin a given digraph Gwill be called simple if the vertex vhas no loop. Otherwise, it will be non-simple.Avertexv∈Vis called source (resp. sink)ifeveryedge which is incident with vertex vhas an orientation from v(resp. toward v). An example of this is represented in Fig. 1. Avertexvin a graph Gis said to be a leaf if it is only adjacent to another vertex. For example, in Fig. 1, all the vertices different from vare leaves. The adjacency matrix of a weighted pseudodigraph G=(V,E)is given by A=(aij), where ai,jis the weight of the edge connecting vertex viwith vj. Obviously, in case that both vertices are not adjacent, that weight would be zero. A sequence of consecutive vertices and edges in a graph is known as a walk.Agraphis connected if there is a walk between any pair of vertices. An arc is a walk where all the edges and vertices are different. The length of a walk is defined by the number of its edges. The length of the shortest walk (also called geodesic) connecting two vertices in a graph Gis the distance between them. The diameter of a graph G,d(G), is the greatest distance between every pair of vertices. Acycle C in a digraph Gis a non-empty walk in which the only repeated vertices are the first and last one. An oriented cycle will be a cycle in the digraph respecting the direction among the directed edges of G. Otherwise, we will call it a non-oriented cycle. A cycle with length kwill be referred as k-cycle. Example 1 In Fig. 2,C:1,a,2,b,3,c,4,d,1 is a non-oriented cycle with length 4, since c is not a directed edge from vertex 3 to 4. For example, C:1,a,2,e,4,d,1isanoriented cycle with length 3. A (directed) tree Tis a (directed) connected graph without cycles. Fig. 1 Sink and source on vertex v, respectively 123 New advances on (pseudo)digraphs and evolution algebras Page 5 of 17 148 Fig. 2 Digraph containing oriented and non-oriented cycles Given two graphs G1=(V1,E1)and G2=(V2,E2), the union of both graphs is defined by G=G1∪G2=(V1∪V2,E1∪E2). If both graphs have no common vertices, we obtain a disconnected graph with components G1and G2. In case that both graphs are joined by a unique common vertex, we will write G1˙ ∪G2. The definition of this operation is similar for digraphs or pseudodigraphs. 3 Method to associate (pseudo)digraphs and evolution algebras In this section, we introduce two different methods. The first one is devoted to obtaining the (pseudo)digraph associated with a given evolution algebra. We also show a few results from paper Ceballos et al. (2020) that will be used later. Finally, we describe the method devoted to defining the evolution algebra associated with a fixed (pseudo)digraph. 3.1 Obtaining the (pseudo)digraph associated with an evolution algebra First, we show how to obtain the (pseudo)digraph associated with a fixed evolution algebra. Let us denote by Ean evolution algebra of dimension n, natural basis B={ei}n i=1and law e2 i=n h=1ci,heh. The pair (E,B)can be associated with a weighted (pseudo)digraph, G, following the method introduced in Ceballos et al. (2020, Section 3), which is as follows: a) For each ei∈B,wedrawvertexi. b) For every ci,i= 0 in the product e2 i(1 ≤i≤n), we draw a loop on vertex iwhose weight is given by ci,i.SeeFig.3. c) For every ci,j= 0(i= j) in the product e2 i(1 ≤i≤n), we draw a directed edge from vertex ito jwhose weight is given by ci,j.Ifcj,i= 0ine2 j, we draw another directed edge, but now from jto iand with weight cj,i.SeeFigs.4and 5. Consequently, every evolution algebra with a natural basis can be associated with a (pseudo)digraph as described in this section. Notice that isolated vertices without loops correspond to basis vectors in the annihilator of the algebra. Let us note that this association is compatible with the one considered in Elduque and Labra (2015) and it is equivalent to considering the structure matrix of Eas the adjacency matrix of G. Example 2 The evolution algebra over the complex number field (from now on, complex evolution algebra) with dimension 4 and non-zero products e1·e1=e2−e3,e2·e2= −e2+e3,e3·e3=e1−e4,e4·e4=e1+e4is associated with the (pseudo)digraph shown in Fig. 6. From now on, we will refer to Ceballos et al. (2020, Propositions 1 and 3). For the convenience of the reader, we include those propositions here. 123 148 Page 6 of 17 M. Ceballos Fig. 3 Loop on i Fig. 4 Directed edge Fig. 5 Double edge Fig. 6 (Pseudo)digraph associated with a 4-dimensional evolution algebra Proposition 1 Let Ebe a 1-dimensional evolution algebra. Then, Eis associated with an isolated vertex with a possible loop. Proposition 2 If a (pseudo)digraph G contains a cycle, then the evolution algebra Eassociated with G is not nilpotent. If an evolution algebra Eis associated with a non-connected (pseudo)digraph G,thenE leads to the direct sum of simple ideals that can be associated with every connected component of G. Bearing this in mind and also Ceballos et al. (2020, Proposition 1), unless it is said, only non-trivial connected (pseudo)digraphs will be considered. It is also necessary to point out the fact that the authors in Ceballos et al. (2020, Proposition 3) should have written “oriented cycle” instead of just “cycle”. 123 New advances on (pseudo)digraphs and evolution algebras Page 7 of 17 148 Fig. 7 Pseudodigraph to be associated with an evolution algebra 3.2 Obtaining the evolution algebra associated with a (pseudo)digraph Now, we see how to define the evolution algebra associated with a fixed (pseudo)digraph. Let G=(V,E)be a pseudodigraph with V={1,...,n}. Then, Gcan be associated with an evolution algebra Ewith natural basis Bas follows: a) Define the vector space W={e1,...,en}from the set of vertices V. b) In case that i(1 ≤i≤n)isasinkvertex,wedefinee2 i=0. c) If iis not a sink, then e2 i=n j=1ci,jej,where ci,jis the weight of the edge from vertex ito vertex j. Notice that, if iis a non-simple vertex, then ci,iwill be the weight of the loop on the vertex i. If there is no directed edge from vertex ito vertex j,thenci,j=0. Example 3 Let us consider the pseudodigraph G=(V,E)with V={1,2,3}represented in Fig. 7. Then, the 3-dimensional evolution algebra Eassociated with Gis the one given by the natural basis B={e1,e2,e3}and products e2 1=e2+e3,e2 3=−2e1+2e2−e3. 4 New results on (pseudo)digraphs and evolution algebras In this section, new results concerning (pseudo)digraphs and evolution algebras are shown. First, we analyze some properties that can be read from the (pseudo)digraph: the annihilator, the derived algebra, and idempotent and absolute nilpotent elements of an evolution algebra. After that, we show several results concerning solvability (or non-solvability), nilpotency, and the preservation of them under the union operation. From here on, Gwill denote the (pseudo)digraph associated with an evolution algebra Eand a fixed natural basis Bfollowing the procedure indicated in Sect. 3. 4.1 Reading properties from the (pseudo)digraph This subsection is devoted to studying which properties of evolution algebras can be read from their associated (pseudo)digraph. Lemma 1 Let us denote by G the (pseudo)digraph associated with an evolution algebra E. Then Ann(E)=span{ei|iis a sink vertex}. Proof It follows from the definition of a sink vertex and the method used in Ceballos et al. (2020, Section 3).  Remark 1 Notice that if Gwere a non-connected (pseudo)digraph associated with E,thenG may contain isolated vertices and we would have to add those corresponding vectors to the 123 148 Page 8 of 17 M. Ceballos Fig. 8 Absolute nilpotent element from non-adjacent vertices basis of Ann(E). Moreover, we would also have to include the vectors associated with sink vertices of each one of its components. Corollary 1 Let G be the (pseudo)digraph associated with E. Then, every sink vertex and each simple isolated vertex corresponds to an absolute nilpotent element of E. Proof It follows from the fact that Ann(E)⊂An(E). Remark 2 Absolute nilpotent elements do not need to be formed from adjacent vertices of G. For example, the 3-dimensional evolution algebra Egiven by e2 1=e2,e2 3=−e2is associated with the digraph of Fig. 8and (e1+e3)2=e2 1+e2 3=0, so e1+e3is absolute nilpotent. Lemma 2 Let G denote the (pseudo)digraph associated with an evolution algebra E.Ifa vertex i of G is non-simple or a source, then the corresponding vector eiis not an absolute nilpotent element of E. Proof If iis a source vertex of G,thene2 i=n j=1ci,jejand ∃1≤k≤n, such that ci,k= 0. Therefore, e2 i= 0. In case that iis a non-simple vertex of G,thenihas a loop with weight ci,i= 0. Consequently, e2 i= 0. In both cases, we obtain ei/∈An(E).  Remark 3 Notice that Lemma 2can be generalized considering a vertex incident with an edge having orientation from it. Lemma 3 Let G be the non-connected (pseudo)digraph associated with an evolution algebra E. Then, every non-simple isolated vertex of G corresponds to an idempotent element of E. Proof Let G=(V,E),whereV={i1,i2,...,in}and ik(1≤k≤n)is an isolated vertex of Ghaving a loop of weight cik,ik= 0. Considering φ:E→Edefined as fik=φ(eik)= 1 cik,ik eik;fij=φ(eij)=eij,∀1≤j= k≤n, we obtain that {fi1,..., fin}is a natural basis of Ewhose associated pseudodigraph has a loop of weight 1 on vertex ikand fikis an idempotent element of E.  Remark 4 Notice that in Lemma 3, we are considering the same natural basis for the pseudodigraph and the idempotent element. Moreover, in the proof of that Lemma, the basis {eih}n h=1and {fih}n h=1are equivalent, since the latter is obtained from the former by multiplying some terms by a non-zero scalar (see Boudi et al. 2022 for more information about equivalent natural basis). Their associated (pseudo)digraphs only differ in the weight of the loops. Lemma 4 Let T be a directed tree. Then, T contains, at least, a sink and a source. 123 New advances on (pseudo)digraphs and evolution algebras Page 9 of 17 148 Fig. 9 Computing the derived algebra for these (pseudo)digraphs Proof As it is well known, Thas, at least, two leaves. If one of those leaves is a sink and another one is a source, then we are done. Now, we assume that all the leaves are sources (the case in which all the leaves are sinks is analogous). Obviously, a leaf cannot have an edge with orientation from it and another edge with orientation toward it due to the own definition of leaf. Let V={v1,...,v n}be the set of vertices of T,where{v1,...,v i}are the leaves. Let Pbe the longest possible walk following the direction of the edges and starting on the leaf v1. We assume that vjis the end of the walk P.If1<j≤i, then the leaf vjis not a source and we obtain a contradiction. If j>i, then there is no edge with orientation from vjdue to the maximality of the walk Pand the fact that Tcontains no cycles. Therefore, vj is a sink vertex.  Lemma 5 Let G the connected (pseudo)digraph associated with E. It is verified that: a) DE=span n h=1ci,heh|iis not a simple sink vertex. b) If G contains no cycles, then Eis not perfect. Proof First, it is trivial that DE=span({e2 i|1≤i≤n})=spann h=1ci,heh.Let us note there is no edge directed from a simple sink vertex. Therefore, we conclude that a) holds. In case that Gcontains no cycle, then Gis a tree. According to Lemma 4,Gcontains at least a source vertex j. Since there is no edge directed to j, we can affirm that ej/∈DE and, hence, Eis not perfect.  Example 4 Let us consider the (pseudo)digraphs Gand Hof Fig. 9.WedenotebyEand Fthe complex evolution algebras associated with Gand H, respectively. Then, Ehas a natural basis {ei}5 i=1with law e2 1=−e2,e2 2=e3+e5,e2 4=2e3.NoticethatGcontains no cycle, so, according to Lemma 5,Eis not perfect. This is true, since DE=span{−e2,2e3,e3+e5}= span{e2,e3,e5}. The evolution algebra Fhas a natural basis {ei}4 i=1, products e2 1=e2−e3,e2 2= −e2−e3,e2 3=e1−e4,e2 4=e1+e4and derived algebra DF=span{e2−e3,−e2− e3,e1+e4,e1−e4}=span{e1,e2,e3,e4}. The pseudodigraph Hcontains cycles and Fis a perfect evolution algebra. 4.2 Solvability and nilpotency In this subsection, we use trees and non-oriented cycles to characterize nilpotent evolution algebras. Next, the cases of solvable non-nilpotent and filiform evolution algebras are shown giving also a correction for a configuration in Ceballos et al. (2020, Proposition 8). Finally, we deal with the union operation studying under which conditions solvability (or non-solvability) and nilpotency are preserved. 123 148 Page 16 of 17 M. Ceballos Fig. 16 Quotients between memory and time Table 2 Number of operations and complexity order of each step Step Procedure Complexity N◦operations 1prod O(n2)N1(n)=1+n(n−1) 2 2diagraph O(n4)N2(n)=n j=1n k=1N1(n) 3diameter O(n4)N3(n)=3+n j=1n k=1N1(n) 7 Conclusions In this paper, new advances concerning the link between evolution algebras and graphs have been shown. Moreover, several elements that can be read from the (pseudo)digraph associated with an evolution algebra have been studied. We have also obtained some results concerning solvability, nilpotency, and their preservation under the graph union operation. Finally, some routines have been implemented to compute the nilpotency index of a nilpotent evolution algebra using its associated digraph. From the author’s point of view, the tools and results shown in this paper may be useful and helpful for understanding the relation between evolution algebras and (pseudo)digraphs. We have seen in the introduction that there are several papers in the literature dealing with the link among evolution algebras and Graph Theory. However, there are several open problems to solve. Here, we have a short list of the problems found in those references and others that come up from this current paper: 1. Study if it is possible to translate each problem in Graph theory to the language of evolution algebras and the reciprocal. This problem was stated by Tian (2008). 2. Find new aspects and properties of evolution algebras that can be read from their associated (pseudo)digraph. 3. Determine which results or elements in graphs are independent of the chosen natural basis of the associated evolution algebras. The last two open problems arise from this current paper and others like Ceballos et al. (2020). The author hopes to deal with them in the near future. 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