The local exponent sets of primitive digraphs

Abstract 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)⩽ exp D (2)⩽⋯⩽ exp D (n)=γ(D) . We define the k th local exponent set E n (k):={ exp D (k)∣D∈PD n } , where PD n is the set of all primitive digraphs of order n . It is known that E n (n)={γ(D)∣D∈PD n } has been completely settled by K. Zhang [Linear Algebra Appl. 96 (1987) 102–108]. In 1998, E n (1) was characterized by J. Shen and S. Neufeld [Linear Algebra Appl. 268 (1998) 117–129]. In this paper, we describe E n (k) for all n,k with 2⩽k⩽n−1 . So the problem of local exponent sets of primitive digraphs is completely solved.