THE SECOND EXPONENT SET OF PRIMITIVE DIGRAPHS

LetD=(V,E) be a primitive digraph. The exponent ofD, denoted by γ(D), is the least integerk such that for anyu, v ∈V there is a directed walk of lengthk fromu tov. The local exponent ofD at a vertexu ∈V, denoted by exp _D (u), is the least integerk such that there is a directed walk of lengthk fromu tov for eachv ∈V. LetV={1,2,...,n}. Following [1], the vertices ofV are ordered so that exp _D (1) ≤exp _D (2) ≤...≤ exp _D (n)=γ(D). LetE _n (i):={exp _D (i) |D∈PD _n }, wherePD _n is the set of all primitive digraphs of ordern. It is known thatE _n (n)={γ(D)|D∈PD _n } has been completely settled by [7]. In 1998,E _n (1) was characterized by [5]. In this paper, the authors describeE _n (2) for alln≥2. Project supported by the National Natural Science Foundation of China and the Jiangsu Provincial Natural Science Foundation of China.