The Generalized Exponent Sets of Primitive, Minimally Strong Digraphs $(I{)}^{\ast }$

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 $V=\{v_1,v_2,\dots ,v_n\}$. The vertices of $V$ can be ordered so that ${\exp }_D(v_{i_1})\le {\exp }_D(v_{i_2})\le \dots \le {\exp }_D(v_{i_n})=\gamma (D)$. 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 $p+q>n$.