  The degree-diameter problem seeks to find the largest possible number of
vertices in a graph having given diameter and given maximum degree. There has
been much recent interest in the problem for mixed graphs, where we allow both
undirected edges and directed arcs in the graph. For a diameter 2 graph with
maximum undirected degree $r$ and directed out-degree $z$, a straightforward
counting argument yields an upper bound $M(z,r,2)=(z+r)^2+z+1$ for the order of
the graph. Apart from the case $r=1$, the only three known examples of mixed
graphs attaining this bound are Cayley graphs, and there are an infinite number
of feasible pairs $(r,z)$ where the existence of mixed Moore graphs with these
parameters is unknown. We use a combination of elementary group-theoretical
arguments and computational techniques to rule out the existence of further
examples of mixed Cayley graphs attaining the Moore bound for all orders up to
485.

Combinatorics