ABSTRACT
Let 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
be the maximum number of nonzero entries of a matrix in
. Given any positive integer t, we prove that
for
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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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.