Designs, Codes and Cryptography

, Volume 68, Issue 1–3, pp 373–393 | Cite as

Codes from incidence matrices of graphs

Article

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.

Keywords

Incidence matrices Graphs Codes Permutation decoding 

Mathematics Subject Classification (2010)

05B05 05C38 94B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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).Google Scholar
  2. 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)MathSciNetMATHCrossRefGoogle Scholar
  3. 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).Google Scholar
  4. 4.
    Björner A., Karlander J.: The mod p rank of incidence matrices for connected uniform hypergraphs. Eur. J. Comb. 14, 151–155 (1993)MATHCrossRefGoogle Scholar
  5. 5.
    Bondy J.A., Murty U.S.R.: Graph Theory with Applications. American Elsevier, New York (1976)MATHGoogle Scholar
  6. 6.
    Brouwer A.E., Haemers W.H.: Eigenvalues and perfect matchings. Linear Algebra Appl. 395, 155–162 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 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).Google Scholar
  8. 8.
    Chartrand G.: A Graph Theoretic Approach to a Communications Problem. SIAM J. Appl. Math. 14, 778–781 (1966)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dankelmann P., Volkmann L.: New sufficient conditions for equality of minimum degree and edge-connectivity. Ars Comb. 40, 270–278 (1995)MathSciNetMATHGoogle Scholar
  10. 10.
    Esfahanian A.H., Hakimi S.L.: On computing a conditional edge-connectivity of a graph. Inform. Process. Lett. 27, 195–199 (1988)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fabrega J., Fiol M.A.: Maximally connected digraphs. J. Graph Theory 13, 657–668 (1989)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fabrega J., Fiol M.A.: Bipartite graphs and digraphs with maximum connectivity. Discret. Appl. Math. 69, 271–279 (1996)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fiol M.A.: On super-edge-connected digraphs and bipartite digraphs. J. Graph Theory 16, 545–555 (1992)MathSciNetMATHCrossRefGoogle Scholar
  14. 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)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fish W., Key J.D., Mwambene E.: Codes from the incidence matrices of graphs on 3-sets. Discret. Math. 311, 1823–1840 (2011)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Fish W., Key J.D., Mwambene E.: Codes from odd graphs (submitted).Google Scholar
  17. 17.
    Ghinelli D., Key J.D.: Codes from incidence matrices and line graphs of Paley graphs. Adv. Math. Commun. 5, 93–108 (2011)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Godsil C., Royle G.: Chromatic number and the 2-rank of a graph. J. Comb. Theory Ser. B 81, 142–149 (2001)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gordon D.M.: Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inform. Theory 28, 541–543 (1982)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hakimi S.L., Bredeson J.G.: Graph theoretic error-correcting codes. IEEE Trans. Inform. Theory 14, 584–591 (1968)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hakimi S.L., Frank H.: Cut-set matrices and linear codes. IEEE Trans. Inform. Theory. 11, 457–458 (1965)CrossRefGoogle Scholar
  22. 22.
    Hellwig A., Volkmann L.: Sufficient conditions for λ′-optimality in graphs of diameter 2. Discret. Math. 283, 113–120 (2004)MathSciNetMATHGoogle Scholar
  23. 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)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Hellwig A., Volkmann L.: Maximally edge-connected and vertex-connected graphs and digraphs—a survey. Discret. Math. 308(15), 3265–3296 (2008)MathSciNetMATHCrossRefGoogle Scholar
  25. 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).Google Scholar
  26. 26.
    Imase M., Nakada H., Peyrat C., Soneoka T.: Sufficient conditions for maximally connected dense graphs. Discret. Math. 63, 53–66 (1987)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Jungnickel D., Vanstone S.A.: Graphical codes—a tutorial. Bull. Inst. Comb. Appl. 18, 45–64 (1996)MathSciNetMATHGoogle Scholar
  28. 28.
    Jungnickel D., Vanstone S.A.: Graphical codes revisited. IEEE Trans. Inform. Theory 43, 136–146 (1997)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Kelmans A.K.: Asymptotic formulas for the probability of k-connectedness of random graphs. Theory Probab. Appl. 17, 243–254 (1972)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Key J.D., Rodrigues B.G.: Codes associated with lattice graphs, and permutation decoding. Discret. Appl. Math. 158, 1807–1815 (2010)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Key J.D., Moori J., Rodrigues B.G.: Permutation decoding for binary codes from triangular graphs. Eur. J. Comb 25, 113–123 (2004)MathSciNetMATHCrossRefGoogle Scholar
  32. 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)MathSciNetMATHCrossRefGoogle Scholar
  33. 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)MathSciNetMATHCrossRefGoogle Scholar
  34. 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)MathSciNetMATHCrossRefGoogle Scholar
  35. 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)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Kroll H.-J., Vincenti R.: PD-sets related to the codes of some classical varieties. Discret. Math 301, 89–105 (2005)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Kroll H.-J., Vincenti R.: Antiblocking systems and PD-sets. Discret. Math 308, 401–407 (2008)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Kroll H.-J., Vincenti R.: Antiblocking decoding. Discret. Appl. Math 158, 1461–1464 (2010)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Li Q.L., Li Q.: Super edge connectivity properties of connected edge symmetric graphs. Networks 33, 157–159 (1999)MathSciNetMATHCrossRefGoogle Scholar
  40. 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).Google Scholar
  41. 41.
    MacWilliams F.J.: Permutation decoding of systematic codes. Bell Syst. Tech. J 43, 485–505 (1964)MATHGoogle Scholar
  42. 42.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1983)Google Scholar
  43. 43.
    Mader W.: Minimale n-fach kantenzusammenängende Graphen. Math. Ann 191, 21–28 (1971)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Plesník J.: Critical graphs of given diameter. Acta Fac. Rerum Nat. Univ. Comenian. Math 30, 79–93 (1975)Google Scholar
  45. 45.
    Plesník J., Znám S.: On equality of edge-connectivity and minimum degree of a graph. Arch. Math. (Brno) 25, 19–25 (1989)MathSciNetMATHGoogle Scholar
  46. 46.
    Schönheim J.: On coverings. Pac. J. Math 14, 1405–1411 (1964)MATHCrossRefGoogle Scholar
  47. 47.
    Shang L., Zhang H.P.: Sufficient conditions for a graph to be λ′-optimal and super-λ′. Networks 49(3), 234–242 (2007)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Tindell R.: Edge-Connectivity Properties of Symmetric Graphs. Stevens Institute of Technology, Hoboken (Preprint) (1982).Google Scholar
  49. 49.
    Volkmann L.: Bemerkungen zum p-fachen zusammenhang von Graphen. An. Univ. Bucuresti Mat 37, 75–79 (1988)MathSciNetMATHGoogle Scholar
  50. 50.
    Wang Y.Q., Li Q.: Super edge-connected properties of graphs with diameter 2. J. Shanghai Jiaotong Univ 33(6), 646–649 (1999)MathSciNetMATHGoogle Scholar
  51. 51.
    Whitney H.: Congruent graphs and the connectivity of graphs. Am. J. Math 54, 154–168 (1932)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Xu J.M.: Restricted edge-connectivity of vertex-transitive graphs. Chin. J. Contemp. Math 21(4), 369–374 (2000)Google Scholar
  53. 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)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  2. 2.School of Mathematical SciencesUniversity of KwaZulu-NatalDurbanSouth Africa

Personalised recommendations