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Articles

0–1 matrices whose k-th powers have bounded entries

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Pages 1972-1982 | Received 04 Apr 2018, Accepted 19 Dec 2018, Published online: 18 Jan 2019
 

ABSTRACT

Let Γ(n,k,t) be the set of 0–1 matrices of order n such that each entry of the k-th powers of these matrices is bounded by t. Let γ(n,k,t) be the maximum number of nonzero entries of a matrix in Γ(n,k,t). Given any positive integer t, we prove that γ(n,k,t)=n(n1)/2 for kn1 when n is sufficiently large, and this maximum number is attained at A if and only if A is permutation similar to the upper triangular tournament matrix of order n.

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2010 Mathematics Subject Classifications:

Acknowledgments

The authors are grateful to the referee for helpful suggestions. The second author also thanks Professor Tin-Yau Tam for helpful discussions on matrix theory during his visit.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Huang was supported by a Fundamental Research Fund for the Central Universities. Partial of this work was done when Huang was visiting Georgia Institute of Technology and Lyu was visiting Auburn University with the financial support of China Scholarship Council.

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