A family of mixed graphs with large order and diameter 2

Abstract

A mixed regular graph is a connected simple graph in which each vertex has both a fixed outdegree (the same indegree) and a fixed undirected degree. A mixed regular graphs is said to be optimal if there is not a mixed regular graph with the same parameters and bigger order.We present a construction that provides mixed graphs of undirected degree q, directed degree q−12 and order 2q2, for q being an odd prime power. Since the Moore bound for a mixed graph with these parameters is equal to 9q2−4q+34 the defect of these mixed graphs is (q−22)2−14.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.