A construction 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 give a construction that provides dense mixed graphs of undirected degree q, directed degree 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 9q 2−4q+3 4 the defect of these mixed graphs is ( q−2 2 ) 2 − 1 4 . In particular we obtain a known mixed Moore graph of order 18, undirected degree 3 and directed degree 1, called Bosák’s graph and a new mixed graph of order 50, undirected degree 5 and directed degree 2, which is proved to be optimal.