Advertisement

Combinatorica

, Volume 8, Issue 2, pp 177–184 | Cite as

Matrices with prescribed row, column and block sums

  • Zh. A. Chernyak
  • A. A. Chernyak
Article

Abstract

The main result of the paper is Theorem 1. It concerns the sets of integral symmetric matrices with given block partition and prescribed row, column and block sums. It is shown that by interchanges preserving these sums we can pass from any two matrices, one from each set, to the other two ones falling “close” together as much as possible. One of the direct corollaries of Theorem 1 is substantiating the fact that any realization ofr-graphical integer-pair sequence can be obtained from any other one byr-switchings preserving edge degrees. This result is also of interest in connection with the problem of determinings-complete properties. In the special cases Theorem 1 includes a number of well-known results, some of which are presented.

AMS subject classifications (1980)

05 B 20 05 C 99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    C. Berge,Graph and Hypergraphs, Dunod, Paris, 1970.Google Scholar
  2. [2]
    D. Billington, Connected subgraphs of the graph of multigraphic realizations of a degree sequences,Lect. Notes Math. 884 (1981), 125–135.Google Scholar
  3. [3]
    R. A. Brualdi, Matrices of zeros and ones with fixed row and column sum vectors,Lin. Al. Appl. 33 (1980), 159–231.Google Scholar
  4. [4]
    M. Capobianco, S. Maurer, D. McCarthy andJ. Molluzo, A collection of open problems,Ann. N.-Y. Acad. Sci. 319 (1979), 565–590.Google Scholar
  5. [5]
    V. Changphaisan, Conditions for sequence to ber-graphic,Discrete Math. 7 (1974), 31–39.Google Scholar
  6. [6]
    Zh. A. Chernyak, Characterization of self-complimentary degree sequences,Doklady Acad. Nayk BSSR 27 (1983), 497–500.Google Scholar
  7. [7]
    Zh. A. Chernyak, Connected realizations of edge degree sequences,Izvestia Akad. Nauk BSSR. Ser. Fiz.-Mat. Nayk 3 (1982), 43–47.Google Scholar
  8. [8]
    Zh. A. Chernyak andA. A. Chernyak, Edge degree sequences and their realizations,Doklady Akad. Nauk BSSR 25 (1981), 594–598.Google Scholar
  9. [9]
    C. J. Colbourn, Graph generation.Dept. of Computer Science, University of Waterloo, Canada, Tech. Report CS-77-37, 1977.Google Scholar
  10. [10]
    P. Das, Unigraphic and Unidigraphic degree sequences through uniquely realizable integer-pair sequences,Discrete Math. 45 (1983), 45–59.Google Scholar
  11. [11]
    R. B. Eggleton andD. A. Holton, Graphic sequences,Lect. Notes Math. 748 (1979), 1–10.Google Scholar
  12. [12]
    D. R. Fulkerson, A. J. Hoffman andM. H. McAndrew, Some properties of graphs with multiple edges,Canad, J. Math. 17 (1965), 166–177.Google Scholar
  13. [13]
    D. J. Kleitman andD. L. Wang, Algorithms for constructing graphs, digraphs with given valences and factors,Discrete Math. 6 (1973), 79–88.Google Scholar
  14. [14]
    S. Kundu, Thek-factor conjecture is true,Discrete Math. 6 (1973), 367–376.Google Scholar
  15. [15]
    A. N. Patrinos andS. L. Hakimi, Relations between graphs and integerpair sequences,Discrete Math. 15 (1976). 347–358.Google Scholar
  16. [16]
    H. J. Ryser, Combinatorial properties of matrices of zeros and ones,Canad. J. Math. 9 (1957), 371–377.Google Scholar
  17. [17]
    R. P. Anstee, The network flows approach for matrices with given row and column sums,Discrete Math. 44 (1983), 125–138.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • Zh. A. Chernyak
    • 1
  • A. A. Chernyak
    • 2
  1. 1.Dept. of Computer TechnicsRadiotechnical InstituteMinskUSSR
  2. 2.Institute of Problems of Machine ReliabilityAcademy of Sciences of BSSRMinskUSSR

Personalised recommendations