The $k$-Exponents of Primitive, Nearly Reducible Matrices

Abstract

Let $D=(V,E)$ be a primitive, minimally strong digraph. In $1982$, J. A. Ross studied the exponent of $D$ and obtained that $\exp (D)\le n+s(n–8)$, where $s$ is the length of a shortest circuit in $D$ $[D]$. In this paper, the $k$-exponent of $D$ is studied. Our principle result is that
$$\begin{gathered} {\exp }_D(k)\le \{\begin{array}{ll}k+1+s(n–3),& \text{if }1\le k\le s,\\ k+s(n-3),& \text{if }s+1\le k\le n,\end{array} \\ . \end{gathered}$$
with equality if and only if $D$ isomorphic to the diagraph $D_{s,n}$ with vertex set $V(D_{s,n})=\{v_1,v_2,\dots ,v_n\}$ and arc set $E(D_{s,n})=\{(v_i,v_{i+1}):1\le i\le n-1\}\cap \{(v_s,v_1),(v_n,v_2)\}$. If $(s,n-1)\ne 1$,then
$${\exp }_D(k)<\{\begin{array}{ll}k+1+s(n–3),& \text{if }1\le k\le s,\\ k+s(n-3),& \text{if }s+1\le k\le n,\end{array}$$ and if $(s,n-1)=1$, then $D_{s,n}$ is a primitive, minimally strong digraph on $n$ vertices with the $k$-exponent $${\exp }_D(k)=\{\begin{array}{ll}k+1+s(n–3),& \text{if }1\le k\le s,\\ k+s(n-3),& \text{if }s+1\le k\le n,\end{array}$$ Moreover, we provide a new proof of Theorem $1$ in $[6]$ and Theorem $2$ in $[14]$ by applying this result.