Matrices with prescribed row, column and block sums


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.

This is a preview of subscription content, access via your institution.


  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 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chernyak, Z.A., Chernyak, A.A. Matrices with prescribed row, column and block sums. Combinatorica 8, 177–184 (1988).

Download citation

AMS subject classifications (1980)

  • 05 B 20
  • 05 C 99