Graphs and nonassociative algebras
type: Item
instance of: scholarly article
title: Graphs and nonassociative algebras
zbMATH Open document ID: 0979.17012
DOI: 10.1007/BF02366345
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-16T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17C10
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 1315224
zbMATH Keywords: graphs
zbMATH Keywords: nonassociative algebras
zbMATH Keywords: Bernstein algebras
Wikidata QID: Q121628200
author: Alexandre N. Grishkov
author: Roberto Costa
reviewed by: A. Castellón Serrano
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1315224