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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

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: EuDML
[1] BALABAN A. T., KEREK F.: Graphs of parallel and or substitution reactions. Rev. Roum de Chemie 19, 1974, No. 4, 631-647.
[2] BERGE C.: Graphs and Hypergraphs. North-Holland, Amsterdam 1973. · Zbl 0483.05029
[3] EGGLETON R. B.: Graphic sequences and graphic polynomials: a report. Infinite and Finite Sets. Colloqu. Math. Soc. J. Bolyai 10, 1973, 385-392.
[4] EGGLETON R. B., HOLTON D. A.: Graphic sequences. Combinatorial Math. VI. Proc. 6th Australian Conf. on Combinatorial Math., Armidale, 1978, Springer-Verlag 1979, 1-10. · Zbl 0425.05051
[5] EGGLETON R. B., HOLTON D. A.: Simple and multigraphic realizations of degree sequences. Combinatorial Math. VIII. Proc. Geelong, Australia 1980, Springer-Verlag 1981, 155-172. · Zbl 0486.05059
[6] ERDÖS P., GALLAI T.: Graphs with prescribed degrees of vertices (in Hungarian). Mat. Lapok 11, 1960, 264-274.
[7] EVANS C. W.: Some properties of semi-regular graphs. match 6, 1979, 117-135. · Zbl 0442.05059
[8] GALE D.: A theorem in folows in networks. Pacific J. Math. 7, 1957, 1073-1082. · Zbl 0087.16303
[9] HAKIMI S. L.: On the realizability of a set of integers as degrees of the vertices of a linear graph. I. J. SIAM 10, 1962, 496-506. · Zbl 0109.16501
[10] HARARY F.: Graph Theory. Addison-Vesley, Mass. 1969. · Zbl 0196.27202
[11] HAVEL V.: A remark on the existence of finite graphs (in Czech). Čas. pěst. mat. 80, 1955, 477-480. · Zbl 0068.37202
[12] MAJCHER Z.: Matrices representable by graphs.: Teubner-Texte zur Mathematik. Band 59, Proc. of the Third Czechoslovak Symposium on Graph Theory, Prague 1982, BSB B. G. Teubner Verlagsgesellschaft, 1983, 178-182.
[13] MAJCHER Z.: On some regularities of graphs II. Čas. p\?st. mat. 109, 1984, 380-388. · Zbl 0576.05058
[14] MAJCHER Z.: Algorithms for constructing paths in a graph of realizations of a degree sequence. · Zbl 0627.05042
[15] PŁONKA J.: On R-regular graphs. Colloquium Math. 46, 1982, 131-134. · Zbl 0496.05047
[16] PŁONKA J.: On some regularities of graphs I. Čas. p\?st. mat. 107, 1982, 231-240. · Zbl 0505.05058
[17] RAMA CHANDRAN S.: Nearly regular graphs and their reconstruction. Graph Theory Newsletter B, 3, 1978.
[18] TAYLOR R.: Constrained switchings in graphs. Combinatorial Math. VIII. Proc. Geelong, Australia 1980, Springer-Verlag 1981, 314-336.
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