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Graphic matrices. (English) Zbl 0662.05045
In the paper finite simple graphs are considered. To every graph G a matrix \(M_ G\) of non-negative integers which informs us about the degrees of the neighbours of each vertex can be assigned. Let \(G=(V,E)\) be a finite simple graph, where \(V=\{v_ 1,...,v_ n\}\) is the vertex set, and let \(D(G)=\{d_ 1,...,d_ k\}\quad (d_ 1>d_ 2>...>d_ k)\) be the sequence of degrees of vertices of V, \(\Gamma (v)=\{u\in V| \quad (u,v)\in E\},\quad V_ i=\{v\in V| \quad \deg v=d_ i\},\quad E_{ij}=\{(u,v)| \quad u\in V_ i,\quad v\in V_ j\},\quad t^ i(v)=| V_ i\cap \Gamma (v)|.\) A (k\(\times n)\)-matrix \(M_ G=(m_{ij})\) with \(m_{ij}=t^ i(v_ j)\) for all \(i\in \{1,...,k\}\), \(j\in \{1,...,n\}\) is called the distribution matrix of G. A matrix M is graphic if \(M=M_ G\) for some graph G. In the paper graphic matrices and the set of all graphs with the same vertex set and with the same distribution matrix is characterized.
Reviewer: V.Fleischer

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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References:
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