For the Kautz digraph $K(d,D)$, let $ρ_k(d,D)$ be the number of oriented edges whose shortest directed cycle has length $k+1$, and define $Δ_k(d,D) = ρ_k(d,D) - ρ_k(d,D-1)$. We give an exact, finite-dimensional matrix product that computes $Δ_k(d,D)$ directly, without first computing $ρ$. In particular, $Δ_k(d,D)=0$ for $k < D/2+2$. and $Δ_k(d,D)$ is positive for every larger $k$ up to $D-1$.
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