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Characterization theorems for the spaces of derivations of evolution algebras associated to graphs. (English) Zbl 1470.17002

The paper studies the space of derivations of \(\mathcal{A}(G)\), an evolution \(K\)-algebra associated to certain type of graph \(G\) (finite, non directed, connected, without multiple edges or loops) with \(\hbox{char}(K)=0\). In this work, a finite graph \(G\) with \(n\) vertices is a pair \((V,E)\) where \(V=\{1,2,\ldots , n\}\) is the set of vertices and \(E:=\{(i,j)\in V\times V \ \colon i\le j\}\) is the set of edges. Two vertices \(i, \ j \in V\) are neighbours if \((i,j)\) or \((j,i) \in E\) and the set of neighbours of a vertex \(i\) is denoted by \(\mathcal{N}(i)\). Two vertices \(i, \ j \in V\) are twins if they have the same set of neighbours. This last definition allows to define the following equivalence relation \(i\sim_t j\) when \(i\) and \(j\) are twins. The evolution algebra associated to \(G\) is the algebra \(\mathcal{A}(G)\) with natural basis \(\{e_i\colon i \in V\}\) and relations \(e_ie_i=\sum_{k \in\mathcal{N}(i)}e_k\) for \(i \in V\) and \(e_ie_j= 0\) if \(i\ne j\). The authors denote by \(\Gamma_3(G)\) the set of all twin classes of \(G\) with at least three vertices and they establish that for a graph \(G\) with \(\vert V \vert \ge 3\) and \(\Gamma_3(G)=\emptyset\), then \( \hbox{Der}(\mathcal{A}(G))=0\). In the case \(\Gamma_3(G)\ne \emptyset\), the authors also give a classification of \(\hbox{Der}(\mathcal{A}(G))\) depending on \(\Gamma_3(G)\).

MSC:

17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
17D92 Genetic algebras
17D99 Other nonassociative rings and algebras
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References:

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