On the exponent of a primitive digraph

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Abstract

We characterize the equality case of the upper bound γ(D) ⪕ n + s(n − 2) for the exponent of a primitive digraph in the case s ⪖ 2, where n is the number of the vertices of the digraph D and s is the length of the shortest elementary circuit of D. We also answer a question about the equality case when s = 1. The exponent of an n × n primitive nonnegative matrix A is the same as the exponent of the associated digraph D(A) of A. So every theorem in this paper can be translated into a theorem about nonnegative matrices.

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This paper forms part of the author's doctoral dissertation written under the supervision of R.A. Brualdi at the University of Wisconsin, 1984.