The local exponent sets of primitive digraphsø

Let D = (V ,E) be a primitive digraph. The local exponent of D at a vertex u ∈ V , denoted by exp D (u), is defined to be the least integer k such that there is a directed walk of length k from u to v for each v ∈ V . Let V = {1, 2, . . ., n}. The vertices of V can be ordered so that exp D (1) 6 exp D (2) 6 · · · 6 exp D (n) = γ (D). We define the kth local exponent set En(k) := {expD(k) | D ∈ PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n) = {γ (D) | D ∈ PDn} has been completely settled by K. Zhang [Linear Algebra Appl. 96 (1987) 102–108]. In 1998, En(1) was characterized by J. Shen and S. Neufeld [Linear Algebra Appl. 268 (1998) 117–129]. In this paper, we describe En(k) for all n, k with 2 6 k 6 n − 1. So the problem of local exponent sets of primitive digraphs is completely solved. © 2000 Published by Elsevier Science Inc. All rights reserved. AMS classification: 05C20; 05C50; 15A33