On locally $n \times n$ grid graphs

TL;DR

This paper studies a special class of grid-like graphs, establishing bounds on their diameter based on local path connectivity, and classifies certain extremal cases with applications to algebraic graph theory.

Contribution

It introduces new bounds on the diameter of locally $n imes n$ grid graphs based on path counts and classifies extremal graphs as distance-regular antipodal covers.

Findings

Diameter bounded by $O( log(n))$ under certain conditions.

Diameter at most 3 if each pair has at least $2(n-1)$ paths.

Infinite family of such graphs for odd prime powers $n$.

Abstract

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 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 $n \times n$ 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 $\mu$-graphs have order at least $8$ or all $\mu$-graphs have order $c$ for some constant $c$.