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Complementary ℓ1-graphs and related combinatorial structures

  • Graph Theory
  • Conference paper
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Combinatorics and Computer Science (CCS 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1120))

Abstract

The conditions under which both graph G and its complement ¯G share some common properties, e.g., with diameter 2, with 1-addressings, with property of being strongly regular are studied. Many examples and counterexamples from different areas of graph theory regarding them are provided. In particular, a census of graphs with at most six vertices is given.

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References

  1. P. Assouad & M. Deza, Espaces metrique plongeables dans un hypercube: aspects combinatoires, Annals of Discrete Mathematics 8(1980) 197–210.

    Google Scholar 

  2. J. Akiyama & F. Harary, A graph and its complement with specified properties 1: connectivity, International Journal of Mathematics and Mathematical Sciences 2(1979) 223–228.

    Article  Google Scholar 

  3. R. Alter, A characterization of self-complementary graphs of order 8, Portugaliae Mathematica 34(1975) 157–161.

    Google Scholar 

  4. A. E. Brouwer, A. M. Cohen, and A. Neumaier, “Distance-Regular Graphs”, Springer-Verlag, 1989.

    Google Scholar 

  5. J. Bosak, “Decompositions of Graphs”, Kluwer Academic Publishers, 1990.

    Google Scholar 

  6. J. Bosak, P. Erdös and A. Rosa, Decompositions of complete graphs into factors with diameter 2, Matematiickj Casopis 21(1971) 14–28.

    Google Scholar 

  7. J. Bosak, A. Rosa and S. Znam, On decomposition of complete graphs into factors with given diameters, in “Theory of Graphs”, P. Erdös, G. Katona (eds.), Academic Press, (1969) 37–56.

    Google Scholar 

  8. D. Cvetković & M. Peric, A table of connected graphs on six verticces, Discrete Mathematics 50(1984) 37–49.

    Article  Google Scholar 

  9. P. K. Das, Almost self-complementary graphs, Ars Combinatoria 31(1991) 267–276.

    Google Scholar 

  10. M. Deza, On the Hamming geometry of unitary cubes, Doklady Akademii Nauk SSR (in Russian) 1037–1040, Soviet Physics Doklady 5(1961) 940–943.

    Google Scholar 

  11. M. Deza and T. Huang, 1-embeddability of some block graphs and cycloids, Bulletin of the Institute of Mathematics, Academia Sinica, to appear (1996).

    Google Scholar 

  12. M. Deza and V.P. Grishukhin, Hypermetric graphs, Quarterly Journal of Oxford (2) (1993), 399–433.

    Google Scholar 

  13. M. Deza and M. Laurent, 1-rigid graphs, Journal of Algebraic Combinatorics 3(1994), 153–175.

    Article  Google Scholar 

  14. M. Deza and H. Maehara, Projecting a simplex onto another one, European Journal of Combinatorics 15(1994), 13–16.

    Article  Google Scholar 

  15. R. C. Entringer, D. E. Jackson and P. A. Snyder, Distance in graphs, Czechoslovak Mathematical Journal 26(1976), 283–296.

    Google Scholar 

  16. R. A. Gibbs, Self-complementary graphs, Journal of Combinatorial Theory, Ser. B, 16(1974) 106–123.

    Google Scholar 

  17. R. L. Graham, B. L. Rothschild, and J. H. Spencer, “Ramsey Theory” 2nd. ed., John-Wiley & Sons (1990).

    Google Scholar 

  18. N. Hartsfield, On regular self-complementary graphs, Journal of Graph Theory 11(1987) 537–538.

    Google Scholar 

  19. D. Hanson, G. MacGillary, On small triangle-free graphs, Ars combinatoria 35(1993) 257–263.

    Google Scholar 

  20. R. Laskar, S. Stueckle and B. Piazza, On the edge-integrity of some graphs and their complements, Discrete Mathematics 122(1993) 245–253.

    Article  Google Scholar 

  21. T. D. Parsons, Orthogonality graphs, Ars Combinatoria 3(1977), 165–208.

    Google Scholar 

  22. T. D. Parsons and T. Pisanki, Exotic n-universal graphs, Journal of Graph Theory 12(1988) 159–167.

    Google Scholar 

  23. I. G. Rosenberg, Regular and strongly regular self-complementary graphs, Annals of Discrete Mathematics 12(1982) 223–238.

    Google Scholar 

  24. S. Ruiz, On strongly regular self-complementary graphs, Journal of Graph Theory 5(1981), 213–215.

    Google Scholar 

  25. S. V. Shpectorov, On scale embeddings of graphs into hypercubes, European Journal of Combinatorics 14(1993) 117–130.

    Article  Google Scholar 

  26. P. Terwilliger and M. Deza, Classification of finite connected hypermetric spaces, Graph and Combinatorics 3(1987), 293–298.

    Article  Google Scholar 

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Michel Deza Reinhardt Euler Ioannis Manoussakis

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© 1996 Springer-Verlag Berlin Heidelberg

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Deza, M., Huang, T. (1996). Complementary ℓ1-graphs and related combinatorial structures. In: Deza, M., Euler, R., Manoussakis, I. (eds) Combinatorics and Computer Science. CCS 1995. Lecture Notes in Computer Science, vol 1120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61576-8_76

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  • DOI: https://doi.org/10.1007/3-540-61576-8_76

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61576-7

  • Online ISBN: 978-3-540-70627-4

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