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 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 (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.

中文翻译：

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在本地 n × n 个网格图


我们在本地研究 n×n 网格图，即任何顶点的邻域是 n 个顶点上两个完整图的笛卡尔积的图。我们考虑这些图的子类，其中距离为 2 的每对顶点都由足够多的长度为 2 的路径连接。根据 Blokhuis 和 Brouwer 之前的工作，已知此类路径的数量最多为 2n。我们表明，如果每对都由至少 2（n-1） 个这样的路径连接，那么直径最多为 3，并且我们在图形的顺序上给出了一个严格的上限。我们表明，满足此上限的图是完整图的距离规则对跖面覆盖。我们展示了一个无限的此类图族，这些图是奇数素数幂 n 的局部 n×n 网格，并将这些结果应用于局部 5×5 网格图，以获得所有μ图（距离 2 处两个顶点的共同邻居集上的诱导子图）至少具有 8 阶或所有 μ 图对于某个常数 c 的阶 c 的情况进行分类。