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Discussiones Mathematicae General Algebra and Applications 24(1) (2004) 53-61
DOI: https://doi.org/10.7151/dmgaa.1075

POWER INDICES OF TRACE ZERO SYMMETRIC BOOLEAN MATRICES

Bo Zhou

Department of Mathematics,
South China Normal University,
Guangzhou 510631, P. R. China
e-mail: zhoubo@scnu.edu.cn

Abstract

The power index of a square Boolean matrix A is the least integer d such that A^d is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n×n primitive symmetric Boolean matrices of trace zero, the class of n×n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zero, and characterize the extreme matrices respectively.
Keywords: power index, index of convergence, period, Boolean matrix.
2000 Mathematics Subject Classification: 15A33, 05C50.

References

[1]	M. Gavalec, Computing matrix period in max-min algebra, Discrete Appl. Math. 75 (1997), 63-70.
[2]	D.A. Gregory, N.J. Pullman and S. Kirkland, On the dimension of the algebra generated by a Boolean matrix, Linear and Multilinear Algebra 38 (1994), 131-144.
[3]	B. Liu, B.D. McKay, N.C. Wormald, and K. Zhang, The exponent set of symmetric primitive (0,1) matrices with zero trace, Linear Algebra Appl. 133 (1990), 121-131.
[4]	S.W. Neufeld, The concept of diameter in exponents of symmetric primitive graphs, Ars Combin. 51 (1999), 129-142.
[5]	G. Ricci, Boolean matrices Ľ neither Boolean nor matrices, Discuss. Math. Gen. Algebra Appl. 20 (2000), 141-151.
[6]	J. Shao, The exponent set of symmetric primitive matrices, Sci. Sinica Ser. A 30 (1987), 348-358.
[7]	J. Shao and Q. Li, On the index of maximum density for irreducible Boolean matrices, Discrete Appl. Math. 21 (1988), 147-156.
[8]	B. Zhou, Exponents of primitive graphs, Australas. J. Combin. 28 (2003), 67-72.

Received 15 July 2003
Revised 12 January 2004