Skip to main content
SpringerLink
Log in

Codes from incidence matrices of graphs

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

We examine the p-ary codes, for any prime p, from the row span over \({\mathbb {F}_p}\) of |V| × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V| or |V| − 1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k − 2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Assmus E.F., Jr., Key J.D.: Designs and Their Codes. Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press, Cambridge (1992) (second printing with corrections, 1993).

  2. Balbuena C., Garcia-Vázquez P., Marcote X.: Sufficient conditions for λ′-optimality in graphs with girth g. J. Graph Theory 52, 73–86 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer D., Boesch F., Suffel C., Tindell R.: Connectivity Extremal Problems and the Design of Reliable Probabilistic Networks. The Theory and Applications of Graphs, Kalamazoo MI, pp. 45–54. Wiley, New York (1981).

  4. Björner A., Karlander J.: The mod p rank of incidence matrices for connected uniform hypergraphs. Eur. J. Comb. 14, 151–155 (1993)

    Article  MATH  Google Scholar 

  5. Bondy J.A., Murty U.S.R.: Graph Theory with Applications. American Elsevier, New York (1976)

    MATH  Google Scholar 

  6. Brouwer A.E., Haemers W.H.: Eigenvalues and perfect matchings. Linear Algebra Appl. 395, 155–162 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cameron P.J., van Lint J.H. : Designs, Graphs, Codes and Their Links. London Mathematical Society Student Texts 22. Cambridge University Press, Cambridge, (1991).

  8. Chartrand G.: A Graph Theoretic Approach to a Communications Problem. SIAM J. Appl. Math. 14, 778–781 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dankelmann P., Volkmann L.: New sufficient conditions for equality of minimum degree and edge-connectivity. Ars Comb. 40, 270–278 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Esfahanian A.H., Hakimi S.L.: On computing a conditional edge-connectivity of a graph. Inform. Process. Lett. 27, 195–199 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fabrega J., Fiol M.A.: Maximally connected digraphs. J. Graph Theory 13, 657–668 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fabrega J., Fiol M.A.: Bipartite graphs and digraphs with maximum connectivity. Discret. Appl. Math. 69, 271–279 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fiol M.A.: On super-edge-connected digraphs and bipartite digraphs. J. Graph Theory 16, 545–555 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fish W., Key J.D., Mwambene E.: Codes from the incidence matrices and line graphs of Hamming graphs. Discret. Math. 310, 1884–1897 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fish W., Key J.D., Mwambene E.: Codes from the incidence matrices of graphs on 3-sets. Discret. Math. 311, 1823–1840 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fish W., Key J.D., Mwambene E.: Codes from odd graphs (submitted).

  17. Ghinelli D., Key J.D.: Codes from incidence matrices and line graphs of Paley graphs. Adv. Math. Commun. 5, 93–108 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Godsil C., Royle G.: Chromatic number and the 2-rank of a graph. J. Comb. Theory Ser. B 81, 142–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gordon D.M.: Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inform. Theory 28, 541–543 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hakimi S.L., Bredeson J.G.: Graph theoretic error-correcting codes. IEEE Trans. Inform. Theory 14, 584–591 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hakimi S.L., Frank H.: Cut-set matrices and linear codes. IEEE Trans. Inform. Theory. 11, 457–458 (1965)

    Article  Google Scholar 

  22. Hellwig A., Volkmann L.: Sufficient conditions for λ′-optimality in graphs of diameter 2. Discret. Math. 283, 113–120 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Hellwig A., Volkmann L.: Sufficient conditions for graphs to be λ′-optimal, super-edge-connected and maximally edge-connected. J. Graph Theory 48, 228–246 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hellwig A., Volkmann L.: Maximally edge-connected and vertex-connected graphs and digraphs—a survey. Discret. Math. 308(15), 3265–3296 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huffman W.C.: Codes and groups. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory Volume 2, Part 2, Chap. 17, pp. 1345–1440. Elsevier, Amsterdam (1998).

  26. Imase M., Nakada H., Peyrat C., Soneoka T.: Sufficient conditions for maximally connected dense graphs. Discret. Math. 63, 53–66 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jungnickel D., Vanstone S.A.: Graphical codes—a tutorial. Bull. Inst. Comb. Appl. 18, 45–64 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Jungnickel D., Vanstone S.A.: Graphical codes revisited. IEEE Trans. Inform. Theory 43, 136–146 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kelmans A.K.: Asymptotic formulas for the probability of k-connectedness of random graphs. Theory Probab. Appl. 17, 243–254 (1972)

    Article  MathSciNet  Google Scholar 

  30. Key J.D., Rodrigues B.G.: Codes associated with lattice graphs, and permutation decoding. Discret. Appl. Math. 158, 1807–1815 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Key J.D., Moori J., Rodrigues B.G.: Permutation decoding for binary codes from triangular graphs. Eur. J. Comb 25, 113–123 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Key J.D., McDonough T.P., Mavron V.C.: Partial permutation decoding for codes from finite planes. Eur. J. Comb. 26, 665–682 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Key J.D., McDonough T.P., Mavron V.C.: Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl 12, 232–247 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. Key J.D., Moori J., Rodrigues B.G.: Codes associated with triangular graphs, and permutation decoding. Int. J. Inform. Coding Theory 1(3), 334–349 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Key J.D., Fish W., Mwambene E.: Codes from the incidence matrices and line graphs of Hamming graphs H k(n,2) for k ≥ 2. Adv. Math. Commun. 5, 373–394 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kroll H.-J., Vincenti R.: PD-sets related to the codes of some classical varieties. Discret. Math 301, 89–105 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kroll H.-J., Vincenti R.: Antiblocking systems and PD-sets. Discret. Math 308, 401–407 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kroll H.-J., Vincenti R.: Antiblocking decoding. Discret. Appl. Math 158, 1461–1464 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li Q.L., Li Q.: Super edge connectivity properties of connected edge symmetric graphs. Networks 33, 157–159 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  40. Liang X., Meng J.: Connectivity of Connected Bipartite Graphs with Two Orbits. Computational Science, ICCS 2007. Lecture Notes in Computer Science, vol. 4489, pp. 334–337. Springer, Berlin (2007).

  41. MacWilliams F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J 43, 485–505 (1964)

    MATH  Google Scholar 

  42. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)

    Google Scholar 

  43. Mader W.: Minimale n-fach kantenzusammenängende Graphen. Math. Ann 191, 21–28 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  44. Plesník J.: Critical graphs of given diameter. Acta Fac. Rerum Nat. Univ. Comenian. Math 30, 79–93 (1975)

    Google Scholar 

  45. Plesník J., Znám S.: On equality of edge-connectivity and minimum degree of a graph. Arch. Math. (Brno) 25, 19–25 (1989)

    MathSciNet  MATH  Google Scholar 

  46. Schönheim J.: On coverings. Pac. J. Math 14, 1405–1411 (1964)

    Article  MATH  Google Scholar 

  47. Shang L., Zhang H.P.: Sufficient conditions for a graph to be λ′-optimal and super-λ′. Networks 49(3), 234–242 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tindell R.: Edge-Connectivity Properties of Symmetric Graphs. Stevens Institute of Technology, Hoboken (Preprint) (1982).

  49. Volkmann L.: Bemerkungen zum p-fachen zusammenhang von Graphen. An. Univ. Bucuresti Mat 37, 75–79 (1988)

    MathSciNet  MATH  Google Scholar 

  50. Wang Y.Q., Li Q.: Super edge-connected properties of graphs with diameter 2. J. Shanghai Jiaotong Univ 33(6), 646–649 (1999)

    MathSciNet  MATH  Google Scholar 

  51. Whitney H.: Congruent graphs and the connectivity of graphs. Am. J. Math 54, 154–168 (1932)

    Article  MathSciNet  Google Scholar 

  52. Xu J.M.: Restricted edge-connectivity of vertex-transitive graphs. Chin. J. Contemp. Math 21(4), 369–374 (2000)

    Google Scholar 

  53. Yuan J., Liu A., Wang S.: Sufficient conditions for bipartite graphs to be super-k-restricted edge connected. Discret. Math 309, 2886–2896 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, 2006, South Africa

    P. Dankelmann

  2. School of Mathematical Sciences, University of KwaZulu-Natal, Durban, 4041, South Africa

    J. D. Key & B. G. Rodrigues

Authors
  1. P. Dankelmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. D. Key
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. G. Rodrigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. D. Key.

Additional information

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dankelmann, P., Key, J.D. & Rodrigues, B.G. Codes from incidence matrices of graphs. Des. Codes Cryptogr. 68, 373–393 (2013). https://doi.org/10.1007/s10623-011-9594-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-011-9594-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation