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An $(r,z,k)$-mixed graph $G$ has every vertex with undirected degree $r$,directed in- and out-degree $z$, and diameter $k$. In this paper, we study thecase $r=z=1$, proposing some new constructions of $(1,1,k)$-mixed graphs with alarge number of vertices $N$. Our study is based on computer techniques forsmall values of $k$ and the use of graphs on alphabets for general $k$. In theformer case, the constructions are either Cayley or lift graphs. In the lattercase, some infinite families of $(1,1,k)$-mixed graphs are proposed withdiameter of the order of $2\log_2 N$.

On large regular (1,1,k)-mixed graphs