We investigate locally n×n grid graphs, that is, graphs in which the
neighbourhood of any vertex is the Cartesian product of two complete graphs on
n vertices. We consider the subclass of these graphs for which each pair of
vertices at distance two is joined by sufficiently many paths of length 2.
The number of such paths is known to be at most 2n by previous work of
Blokhuis and Brouwer. We show that if each distance two pair is joined by at
least n−1 paths of length 2 then the diameter is bounded by O(log(n)),
while if each pair is joined by at least 2(n−1) such paths then the diameter
is at most 3 and we give a tight upper bound on the order of the graphs. We
show that graphs meeting this upper bound are distance-regular antipodal covers
of complete graphs. We exhibit an infinite family of such graphs which are
locally n×n grid for odd prime powers n, and apply these results to
locally 5×5 grid graphs to obtain a classification for the case where
either all μ-graphs have order at least 8 or all μ-graphs have order
c for some constant c