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Enumeration of Graphs with given Partition

  • K. R. Parthasarathy (a1)

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In this paper we use a generalized form of Polya's theorem (1) to obtain generating functions for the number of ordinary graphs with given partition and for the number of bicoloured graphs with given bipartition. Both the points and lines of the graphs are taken as unlabelled. These graph enumeration problems were proposed by Harary in his review article (4). Read (7, 8) solved the problem for unlabelled general graphs and labelled ordinary graphs.

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References

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1. de Bruijn, N. G. Generalization of Polya1 s fundamental theorem in enumerative combinatorial analysis, Indag. Math., 21 (1959), 5969.
2. Harary, F., The number of linear, directed, rooted and connected graphs, Trans. Amer. Math. Soc, 78 (1955), 445463.
3. Harary, F., On the number of bicoloured graphs, Pacific J. Math., 8 (1958), 743755.
4. Harary, F., Unsolved problems in the enumeration of graphs, Publ. Math. Inst. Hung. Acad. Sci., 5 (1960), 6395.
5. McMahon, P. A., Combinatory analysis, Vol. I (Cambridge 1915; New York 1960).
6. Mirsky, L., Inequalities and existence theorems in the theory of matrices, J. Math. Anal, and Appl., 9 (1964), 99118.
7. Read, R. C., The enumeration of locally restricted graphs I, J. London Math. Soc, 84 (1959), 417436.
8. Read, R. C., The enumeration of locally restricted graphs II, J. London Math. Soc, 85 (1960), 344351.
9. Ryser, H. J., Matrices of zeros and ones, Bull. Amer. Math. Soc, 66 (1960), 442464.
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