Number of edges with shortest cycle k in a Kautz graph

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