New cyclic Kautz digraphs with optimal diameter

We obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and out-degree, there is no other digraph with a smaller diameter. This new family of digraphs are called `modified cyclic digraphs' $MCK(d,\ell)$, and it is derived from the Kautz digraphs $K(d,\ell)$ and from the so-called cyclic Kautz digraphs $CK(d,\ell)$. The cyclic Kautz digraphs $CK(d,\ell)$ were defined as the digraphs whose vertices are labeled by all possible sequences $a_1\ldots a_\ell$ of length $\ell$, such that each character $a_i$ is chosen from an alphabet of $d+1$ distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and also requiring that $a_1\neq a_\ell$. Their arcs are between vertices $a_1 a_2\ldots a_\ell$ and $a_2 \ldots a_\ell a_{\ell+1}$, with $a_1\neq a_\ell$ and $a_2\neq a_{\ell+1}$. Since $CK(d,\ell)$ do not have minimal diameter for their number of vertices, we construct the modified cyclic Kautz digraphs to obtain the same diameter as in the Kautz digraphs, and we also show that $MCK(d,\ell)$ are $d$-out-regular. Moreover, for $t\geq1$, we compute the number of vertices of the iterated line digraphs $L^t(CK(d,\ell))$