The Generalized Exponent Sets of Primitive, Minimally Strong Digraphs \((I)^*\)

Abstract

Let \(D = (V, E)\) be a primitive digraph. The exponent of \(D\) at a vertex \(u \in V\), denoted by \(\exp_D(u)\), is defined to be the least integer \(k\) such that there is a walk of length \(k\) from \(u\) to \(v\) for each \(v \in V\). Let \(V = \{v_1, v_2, \ldots, v_n\}\). The vertices of \(V\) can be ordered so that \(\exp_D(v_{i_1}) \leq \exp_D(v_{i_2}) \leq \ldots \leq \exp_D(v_{i_n}) = \gamma(D)\). The number \(\exp_p(v_n)\) is called the \(k\)-exponent of \(D\), denoted by \(\exp_p(k)\). We use \(L(D)\) to denote the set of distinct lengths of the cycles of \(D\). In this paper, we completely determine the \(1\)-exponent sets of primitive, minimally strong digraphs with \(n\) vertices and \(L(D) = \{p, q\}\), where \(3 \le p < q\) and \(p + q > n\).