Approaching the mixed Moore bound for diameter two by Cayley graphs

In a mixed (Δ, d)-regular graph, every vertex is incident with Δ ≥ 1 undirected edges and there are d ≥ 1 directed edges entering and leaving each vertex. If such a mixed graph has diameter 2, then its order cannot exceed (Δ+ d) + d+1. This quantity generalizes the Moore bounds for diameter 2 in the case of undirected graphs (when d = 0) and digraphs (when Δ = 0). For every d such that d − 1 is a prime power, Kautz digraphs of inand out-degree d are Cayley digraphs of order missing the directed Moore bound by just 1. At the other extreme, the author and J. Širáň (2012) proved that the undirected Moore bound for diameter 2 and degree Δ can be asymptotically approached by Cayley graphs for an infinite set of values of Δ. We consider extensions of these results to mixed Cayley graphs, that is, mixed (Δ, d)-regular graphs admitting a group of automorphisms acting regularly on vertices.