On locally n × n grid graphs

We investigate locally $n\times n$ 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 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 $n\times n$ grid for odd prime powers n, and apply these results to locally $5\times 5$ grid graphs to obtain a classification for the case where either all μ-graphs (induced subgraphs on the set of common neighbours of two vertices at distance two) have order at least 8 or all μ-graphs have order c for some constant c.