The Extremal Primitive Digraph with Both Lewin Index $n–2$ and Girth $2$ or $3$

Abstract

Let $D$ be a primitive digraph. Then there exists a nonnegative integer $k$ such that there are walks of length $k$ and $k+1$ from $u\$to\text{(}v$ for some $u,v\in V(D)$ (possibly $u$ again ). Such smallest $k$ is called the Lewin index of the digraph $D$, denoted by $l(D)$. In this paper, the extremal primitive digraphs with both Lewin index $n—2$ and girth $2$ or $3$ are determined.