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Generalized competition index of primitive digraphs

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Abstract

For any positive integers k and m, the k-step m-competition graph C k m (D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v1, v2, · · ·, v m in D such that there are directed walks of length k from x to v i and from y to v i for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C k m (D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.

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References

  1. Akelbek, M., Kirkland, S. Coefficients of ergodicity and the scrambling index. Linear Algebra Appl., 430: 1111–1130 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Akelbek, M., Kirkland, S. Primitive digraphs with the largest scrambling index. Linear Algebra Appl., 430: 1099–1110 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Akelbek, M., Fital, S., Shen, J. A bound on the scrambling index of a primitive matrix using Boolean rank. Linear Algebra Appl., 431: 1923–1931 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brualdi, R.A., Ryser, H.J. Combinatorial Matrix Theory. Cambridge University Press, Cambridge, 1991

    Book  MATH  Google Scholar 

  5. Bondy, J.A., Murty, U.S.R. Graph Theory with Applications. North-Holland, New York, 1976

    Book  MATH  Google Scholar 

  6. Cho, H.H., Kim, S.R., Nam, Y. The m-step competition graph of digraph. Discrete Appl. Math., 105: 115–127 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cho, H.H., Kim, H.K. Competition indices of digraphs. In: Proceedings of Workshop in Combinatorices, 2004, 99–107

    Google Scholar 

  8. Chen, S.X., Liu, B.L. The scrambling index of symmetric primitive matrices. Linear Algebra Appl., 433: 1110–1126 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cohen, J.E. Food Webs and Niche Space. Princeton University Press, Princeton, NJ,1978

    Google Scholar 

  10. Huang, Y.F., Liu, B.L. Generalized scrambling indices of a primitive digraph. Linear Algebra Appl., 433: 1798–1808 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kim, H.K. Competition indices of tournaments. Bull. Korean Math. Soc., 45: 385–396 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kim, H.K. Generalized competition index of a primitive digraph. Linear Algebra Appl., 433: 72–79 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim, H.K. A bound on the generalized competition index of a primitive matrix using Boolean rank. Linear Algebra Appl., 435: 2166–2174 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, H.K., Lee, S.H. Generalized competition indices of symmetric primitive digraphs. Discrete Appl. Math., 160: 1583–1590 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kim, H.K., Park, S.G. A bound of generalized competition index of a primitive digraph. Linear Algebra Appl., 436: 86–98 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kim, H.K. Generalized competition index of an irreducible Boolean matrix. Linear Algebra Appl., 438: 2747–2756 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kim, H.K. Scrambling index set of primitive digraphs. Linear Algebra Appl., 439: 1886–1893 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, H.K. Characterization of irreducible Boolean matrices with the largest generalized competition index. Linear Algebra Appl., 466: 218–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, S.R. Competition graphs and scientific laws for food webs and other systems. PH.D. Thesis, Rutgers University, 1988

    Google Scholar 

  20. Liu, B.L. On fully indecoposable exponent for primitive Boolean matrices symmetric ones. Linear Multilinear Algebra., 31: 131–138 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, B.L., Lai, H.J. Matrices in Combinatorics and Graph Theory. Kluwer Academic Publishers, 2000

    Book  MATH  Google Scholar 

  22. Liu, B.L., Huang, Y.F. The scrambling index of primitive digraphs. Comput. Math. Appl., 60: 706–721 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moon, J.W., Moser, L. Almost all (0, 1)-matrices are primitive. Studia Scient. Math. Hung., 1: 153–156 (1966)

    MathSciNet  MATH  Google Scholar 

  24. Shao, J.Y. On the exponent set of symmetric primitive matrices. Sci. Sinica Ser. A., 9: 931–939 (1986)

    Google Scholar 

  25. Shao, Y.L., Gao, Y.B. The primitive Boolean matrices with the second largest scrambling index by Boolean rank. Czechoslovak Mathematical J., 64 (139): 269–283 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shao, Y.L., Gao, Y.B. The scrambling index set of primitive minimally strong digraphs. Linear Algebra Appl., 500: 1–14 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Shen, J., Gregory, D., Neufeld, S. Exponents of indecomposability. Linear Algebra Appl., 288: 229–241 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Sim, M.S., Kim, H.K. On generalized competition index of a primitive tournament. Discrete Math., 311: 2657–2662 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. You, L.H., Liu, B.L., Shen, J. r-indecomposable and r-nearly decomposable matrices. Linear Algebra Appl., 407: 105–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang L., Huang, T.Z. Bounds on the generalized µ-scrambling indices of primitive digraphs. International Journal of Computer Mathematics, 89: 17–29 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhou, B. On the common consequent indices of nearly reducible binary relations. Util. Math., 56: 233–243 (1999)

    MathSciNet  MATH  Google Scholar 

  32. Zhou, B., Liu, B.L. New results on the common consequent index of a binary relation. European J. Combin., 21: 167–179 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Li-hua You & Bo Zhou

  2. Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA

    Fang Chen & Jian Shen

Authors
  1. Li-hua You
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  2. Fang Chen
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  3. Jian Shen
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  4. Bo Zhou
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Corresponding author

Correspondence to Li-hua You.

Additional information

Supported by the National Natural Science Foundation of China (No. 11571123, 11671156) and the Guangdong Provincial Natural Science Foundation (Grant No. 2015A030313377).

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You, Lh., Chen, F., Shen, J. et al. Generalized competition index of primitive digraphs. Acta Math. Appl. Sin. Engl. Ser. 33, 475–484 (2017). https://doi.org/10.1007/s10255-017-0675-0

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  • DOI: https://doi.org/10.1007/s10255-017-0675-0

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