Abstract
A digraph G is called primitive if for some positive integer k, there is a walk of length exactly k from each vertex u to each vertex v (possibly u again). If G is primitive, the smallest such k is called the exponent of G, denoted by exp(G). For any real number r, 0<r<1, let f(n,r) be the maximum number of arcs in a primitive digraph with n vertices having exponent greater than or equal to r2n2. We show that f(n,r)/n2 is asymptotically (1−r)2/3 whenever r⩾2/2.
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