The local exponent sets of primitive digraphs Original Research Article

Let D=(V,E) be a primitive digraph. The local exponent of D at a vertex uset membership, variantV, denoted by expD(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 vset membership, variantV. Let V={1,2,…, n}. The vertices of V can be ordered so that expD(1)less-than-or-equals, slantexpD(2)less-than-or-equals, slantcdots, three dots, centeredless-than-or-equals, slantexpD(n)=γ(D). We define the kth local exponent set En(k):={expD(k)midDset membership, variantPDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n)={γ(D)midDset membership, variantPDn} 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 2less-than-or-equals, slantkless-than-or-equals, slantn−1. So the problem of local exponent sets of primitive digraphs is completely solved.