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 \(\exp(D) \leq n + s(n – 8)\), 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
\[
\exp_D(k) \leq \begin{cases}
k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\
k + s(n-3), & \text{if } s+1 \leq k \leq n,\\
\end{cases} \\.
\]
with equality if and only if \(D\) isomorphic to the diagraph \(D_{s,n}\) with vertex set \(V(D_{s,n})=\{v_1,v_2,\ldots,v_n\}\) and arc set \(E(D_{s,n})=\{(v_i,v_{i+1}):1\leq i\leq n-1\}\cap \{(v_s,v_1),(v_n,v_2)\}\). If \((s,n-1)\neq 1\),then
\[
\exp_D(k)< \begin{cases} k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\ k + s(n-3), & \text{if } s+1 \leq k \leq n, \end{cases} \\ \] and if \((s,n-1)=1\), then \(D_{s, n}\) is a primitive, minimally strong digraph on \(n\) vertices with the \(k\)-exponent \[ \exp_D(k)= \begin{cases} k + 1 + s(n – 3), & \text{if } 1 \leq k \leq s, \\\ k + s(n-3), & \text{if } s+1 \leq k \leq n, \end{cases} \\ \] Moreover, we provide a new proof of Theorem \(1\) in \([6]\) and Theorem \(2\) in \([14]\) by applying this result.