  One of the prominent areas of research in graph theory is the degree-diameter
problem, in which we seek to determine how many vertices a graph may have when
constrained to a given degree and diameter. Different variants of this problem
are obtained by considering further restrictions on the graph, such as whether
it is directed, undirected, mixed, vertex-transitive etc. The currently best
known extremal constructions are the G\'omez graphs, both when considered as
directed and undirected. As the G\'omez graphs are similar to the
Faber-Moore-Chen graphs, we give a natural generalisation of their definition
which we call word graphs. We then prove that the G\'omez graphs are as large
as possible word graphs for given degree and diameter. Further, we provide a
test to determine the full automorphism group of a word graph, and apply it to
the extremal directed G\'omez graphs to settle the previously open question of
when the G\'omez graphs are also Cayley graphs. Finally, we conclude with a
brief list of interesting related open problems.
