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A bound on the exponent of a primitive matrix using Boolean rank

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
217
Identifiers
DOI: 10.1016/0024-3795(92)00003-5

Abstract

Abstract We present a bound on the exponent exp( A) of an n × n primitive matrix A in terms of its boolean rank b = b( A); namely exp( A) ≤ ( b − 1) 2 + 2. Further, we show that for each 2 ≤ b ≤ n − 1, there is an n × n primitive matrix A with b( A) = b such that exp( A) = ( b − 1) 2 + 2, and we explicitly describe all such matrices. The new bound is compared with a well-known bound of Dulmage and Mendelsohn, and with a conjectured bound of Hartwig and Neumann. Several open problems are posed.

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