Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ENUMERATION OF GRAPHS WITH GIVEN WEIGHTED NUMBER OF CONNECTED COMPONENTS

  • Received : 2015.11.10
  • Accepted : 2017.06.01
  • Published : 2017.11.30
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Abstract

We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of q-partite graphs of given order, size and number of connected components.

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References

  1. G. W. Ford, R. Z. Norman, and G. E. Uhlenbeck, Combinatorial problems in the theory of graphs. II, Proc. Nat. Acad. Sci. U. S. A. 42 (1956), 203-208.
  2. P. Hanlon, The enumeration of bipartite graphs, Discrete Math. 28 (1979), no. 1, 49-57. https://doi.org/10.1016/0012-365X(79)90184-5
  3. F. Harary, On the number of bi-colored graphs, Pacific J. Math. 8 (1958), 743-755. https://doi.org/10.2140/pjm.1958.8.743
  4. F. Harary and E. M. Palmer, Graphical enumeration, Academic Press, New York- London, 1973.
  5. F. Harary and G. Prins, Enumeration of bicolourable graphs, Canad. J. Math. 15 (1963), 237-248. https://doi.org/10.4153/CJM-1963-028-2
  6. T. W. Melnyk, J. S. Rowlinson, and B. L. Sawford, The penetrable sphere and related models of liquid vapour equilibrium, Molecular Physics 24 (1972), no. 4, 809-831. https://doi.org/10.1080/00268977200101921
  7. R. C. Read, Some enumeration problems in graph theory, Doctoral Thesis, University of London, 1958.
  8. R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math. 12 (1960), 410-414. https://doi.org/10.4153/CJM-1960-035-0
  9. J. Song, Characteristic polynomial of certain hyperplane arrangements through graph theory, (submitted for publication), arXiv:1701.07330 [math.CO].
  10. J. Song, Enumeration of graphs and the characteristic polynomial of the hyperplane arrangements Jn, J. Korean Math. Soc. (to appear), arXiv:1701.07313 [math.CO].
  11. J. Song, On certain hyperplane arrangements and colored graphs, Bull. Korean Math. Soc. 54 (2017), no. 2, 375-382. https://doi.org/10.4134/BKMS.b150167
  12. R. P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999.
  13. M. Ueno and S. Tazawa, Enumeration of bipartite self-complementary graphs, Graphs Combin. 30 (2014), no. 2, 471-477. https://doi.org/10.1007/s00373-012-1275-7