  The undirected degree/diameter and degree/girth problems and their directed
analogues have been studied for many decades in the search for efficient
network topologies. Recently such questions have received much attention in the
setting of mixed graphs, i.e. networks that admit both undirected \emph{edges}
and directed \emph{arcs}. The degree/diameter problem for mixed graphs asks for
the largest possible order of a network with diameter $k$, maximum undirected
degree $\leq r$ and maximum directed out-degree $\leq z$. It is also of
interest to find smallest possible $k$-geodetic mixed graphs with minimum
undirected degree $\geq r$ and minimum directed out-degree $\geq z$. A simple
counting argument reveals the existence of a natural bound, the \emph{Moore
bound}, on the order of such graphs; a graph that meets this limit is a
\emph{mixed Moore graph}. Mixed Moore graphs can exist only for $k = 2$ and
even in this case it is known that they are extremely rare. It is therefore of
interest to search for graphs with order one away from the Moore bound. Such
graphs must be out-regular; a much more difficult question is whether they must
be totally regular. For $k = 2$, we answer this question in the affirmative,
thereby resolving an open problem stated in a recent paper of L\'opez and
Miret. We also present partial results for larger $k$. We finally put these
results to practical use by proving the uniqueness of a 2-geodetic mixed graph
with order exceeding the Moore bound by one.
