COMPETITION INDICES OF TOURNAMENTS

Title & Authors
COMPETITION INDICES OF TOURNAMENTS
Kim, Hwa-Kyung;

Abstract
For a positive integer m and a digraph D, the m-step competition graph $\small{C^m}$ (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.
Keywords
competition graph;m-step competition graph;competition index;competition period;tournament;
Language
English
Cited by
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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

3.
R. A. Brualdi and J. Shao, Generalized exponents of primitive symmetric digraphs, Discrete Appl. Math. 74 (1997), 275-293

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

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

10.
B. R. Heap and M. S. Lynn, The structure of powers of nonnegative matrices, SIAM J. Appl. Math. 14 (1966), 610-640

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

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

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

18.
B. Zhou and J. Shen, On generalized exponents of tournaments, Taiwanese J. Math. 6 (2002), 565-572