A Family of Dense Mixed Graphs of Diameter $2$

A mixed graph is said to be dense if its order is close to the Moore bound and it is optimal if there is not a mixed graph with the same parameters and bigger order. We present a construction that provides dense mixed graphs of undirected degree $q$, directed degree $\frac{q-1}{2}$ and order $2q^2$, for $q$ being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to $\frac{9q^2-4q+3}{4}$ the defect of these mixed graphs is $({\frac{q-2}{2}})^2-\frac{1}{4}$. In particular we obtain a known mixed Moore graph of order $18$, undirected degree $3$ and directed degree $1$ called Bos\'ak's graph and a new mixed graph of order $50$, undirected degree $5$ and directed degree $2$, which is proved to be optimal.