# COMPETITION INDICES OF TOURNAMENTS

• Kim, Hwa-Kyung (Department of Mathematics Education Sangmyung University)
• Published : 2008.05.31

#### Abstract

For a positive integer m and a digraph D, the m-step competition graph $C^m$ (D) of D has he same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that there are directed walks of length m from u to x and from v to x. Cho and Kim [6] introduced notions of competition index and competition period of D for a strongly connected digraph D. In this paper, we extend these notions to a general digraph D. In addition, we study competition indices of tournaments.

#### References

1. R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, 1991
2. R. A. Brualdi and B. L. Liu, Generalized exponents of primitive directed graphs, J. Graph Theory 14 (1991), 483-499 https://doi.org/10.1002/jgt.3190140413
3. R. A. Brualdi and J. Shao, Generalized exponents of primitive symmetric digraphs, Discrete Appl. Math. 74 (1997), 275-293 https://doi.org/10.1016/S0166-218X(96)00077-7
4. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, New York, 1976
5. H. H. Cho, Indices of irreducible Boolean matrix, J. Korean Math. Soc. 30 (1993), 78-85
6. H. H. Cho and H. K. Kim, Competition indices of digraphs, Proceedings of workshop in combinatorics (2004), 99-107
7. H. H. Cho, S.-R. Kim, and Y. Nam, The m-step competition graph of a digraph, Discrete Appl. Math. 105 (2000), 115-127 https://doi.org/10.1016/S0166-218X(00)00214-6
8. J. E. Cohen, Food Webs and Niche Space, Princeton Univ. Press, Princeton, NJ, 1978
9. H. J. Greenberg, J. R. Lundgren, and J. S. Maybee, Inverting graphs of rectangular matrices, Discrete Appl. Math. 8 (1984), 255-265 https://doi.org/10.1016/0166-218X(84)90123-9
10. B. R. Heap and M. S. Lynn, The structure of powers of nonnegative matrices, SIAM J. Appl. Math. 14 (1966), 610-640 https://doi.org/10.1137/0114052
11. S.-R. Kim, Competition graphs and scientific laws for food webs and other systems, Ph. D. Thesis, Rutgers University, 1988
12. B. Liu and H. J. Lai, Matrices in Combinatorics and Graph Theory, Kluwer Academic Publishers, 2000
13. J. W. Moon and N. J. Pullman, On the power of tournament matrices, J. Combinatorial Theory 3 (1967), 1-9 https://doi.org/10.1016/S0021-9800(67)80009-7
14. J. Shao, The exponent set of symmetric primitive matrices, Scientia Sinica, Ser. A 30 (1987), 348-358
15. J. Shao and S.-G. Hwang, Generalized exponents of non-primitive graphs, Linear Algebra Appl. 279 (1998), 207-225 https://doi.org/10.1016/S0024-3795(98)10014-9
16. J. Shao and Q. Li, The indices of convergence reducible Boolean matrices, Acta Math. Sinica 33 (1990), 13-28
17. J. Shen, Proof of a conjecture about the exponent of primitive matrics, Linear Algebra and Its Appl. 216 (1995), 185-203 https://doi.org/10.1016/0024-3795(93)00132-J
18. B. Zhou and J. Shen, On generalized exponents of tournaments, Taiwanese J. Math. 6 (2002), 565-572 https://doi.org/10.11650/twjm/1500407480

#### Cited by

1. COMPETITION INDICES OF STRONGLY CONNECTED DIGRAPHS vol.48, pp.3, 2011, https://doi.org/10.4134/BKMS.2011.48.3.637
2. Generalized competition indices of symmetric primitive digraphs vol.160, pp.10-11, 2012, https://doi.org/10.1016/j.dam.2012.03.001
3. Generalized competition index of primitive digraphs vol.33, pp.2, 2017, https://doi.org/10.1007/s10255-017-0675-0
4. A matrix sequence{Γ(Am)}m=1∞might converge even if the matrix A is not primitive vol.438, pp.5, 2013, https://doi.org/10.1016/j.laa.2012.10.012
5. THE COMPETITION INDEX OF A NEARLY REDUCIBLE BOOLEAN MATRIX vol.50, pp.6, 2013, https://doi.org/10.4134/BKMS.2013.50.6.2001
6. On generalized competition index of a primitive tournament vol.311, pp.23-24, 2011, https://doi.org/10.1016/j.disc.2011.08.012
7. Generalized scrambling indices of a primitive digraph vol.433, pp.11-12, 2010, https://doi.org/10.1016/j.laa.2010.06.043
8. A bound of generalized competition index of a primitive digraph vol.436, pp.1, 2012, https://doi.org/10.1016/j.laa.2011.06.040
9. Generalized competition index of a primitive digraph vol.433, pp.1, 2010, https://doi.org/10.1016/j.laa.2010.01.033
10. Bounds on the generalized μ-scrambling indices of primitive digraphs vol.89, pp.1, 2012, https://doi.org/10.1080/00207160.2011.638059
11. On the matrix sequence for a Boolean matrix A whose digraph is linearly connected vol.450, 2014, https://doi.org/10.1016/j.laa.2014.02.046
12. Characterization of irreducible Boolean matrices with the largest generalized competition index vol.466, 2015, https://doi.org/10.1016/j.laa.2014.10.004
13. The scrambling index of primitive digraphs vol.60, pp.3, 2010, https://doi.org/10.1016/j.camwa.2010.05.018
14. Generalized competition index of an irreducible Boolean matrix vol.438, pp.6, 2013, https://doi.org/10.1016/j.laa.2012.10.040
15. A bound on the generalized competition index of a primitive matrix using Boolean rank vol.435, pp.9, 2011, https://doi.org/10.1016/j.laa.2011.04.002
16. Scrambling index set of primitive digraphs vol.439, pp.7, 2013, https://doi.org/10.1016/j.laa.2013.05.022