Graphs and nonassociative algebras (Q1296172)

Given a graph \(G=(V,E)\), where \(V\) is the set of vertices and \(E\) the set of edges, \textit{R. Costa} and \textit{H. Guzzo jun.} [Commun. Algebra 25, 2129-2139 (1997; Zbl 0879.17016)] have constructed a nonassociative algebra \(A(G)\) over a field \(K\) associated with the graph \(G\). Concretely, \(A(G)=U\oplus Z\), with \(U=\oplus_{v\in V}Kv\), \(Z=\oplus_{(a,b)\in S}Kz_a^b\), and \(z_a^b\) is the linear operator in \(U\) defined by the rules: (i) \(z_a^b=z_b^a\), for any \(a,b\in V\), and (ii) \(z_a^b(b)=a\), \(z_a^b(a)=b\), \(z_a^b(c)=0\), for any \(a,b,c\in V\), \(c\notin\{ a,b\}\). The product in \(A(G)\) is introduced by \((x\oplus f)(y\oplus g)=g(x)\oplus f(y)\). If \(K\) has characteristic not two, \(Ke\oplus A(G)\) is endowed with the structure of a Bernstein algebra by means of \(2ev=v\) and \(ez_a^b=0\) (\(v,a,b\in V\)). In this paper, it is proved that if \(G\) and \(G'\) are simple connected graphs such that the algebras \(A(G)\) and \(A(G')\) over the field \(K\) of characteristic \(\neq 2\) are isomorphic, then the graphs \(G\) and \(G'\) are isomorphic.