Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u --> v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u E V, denoted exp(u), is the least integer k such that there is a u --> v walk of length k for each v is an element of V. For a set X subset of or equal to V, exp(X) is the least integer k such that for each v is an element of V there is a X --> v walk of length k, i.e., a u --> v walk of length k for some u is an element of X. Let F(G, k) := max{exp(X) : \X\ = k} and F(n, k) := max{F(G, k) : \V\ = n}, where \X\ and \V\ denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n - k) (n - 1) + 1 for all 1 less than or equal to k less than or equal to n - 1. In this article, for each k, 1 less than or equal to k less than or equal to n - 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Delta(+)(G) = max{d(+)(u) : u is an element of V}, and thus obtain a new refinement of the Wielandt bound (n - 1)(2) + 1. (C) 1998 John Wiley & Sons, Inc.