Let \(\varDelta\) be a digraph of diameter 2 with the maximum undirected vertex degree t and the maximum directed out-degree z. The largest possible number v of vertices of \(\varDelta\) is given by the following generalization of the Moore bound:

and a digraph attaining this bound is called a Moore digraph. Apart from the case \(t=1\), only three Moore digraphs are known, which are also Cayley graphs. Using computer search, Erskine (J Interconnect Netw, 17: 1741010, 2017) ruled out the existence of further examples of Cayley digraphs attaining the Moore bound for all orders up to 485. We use an algebraic approach to this problem, which goes back to an idea of G. Higman from the theory of association schemes, also known as Benson’s Lemma in finite geometry, and show non-existence of Moore Cayley digraphs of certain orders.

令\（\ varDelta \）是直径2的有向图，其最大无向顶点度为t，最大有向度为z。\（\ varDelta \）的最大可能顶点数v由以下摩尔定律的一般化给出：

达到此界限的有向图称为摩尔有向图。除了情况\（t = 1 \）外，仅知道三个Moore有向图，它们也是Cayley图。通过使用计算机搜索，Erskine（J Interconnect Netw，17：1741010，2017）排除了存在更多Cayley有向图的例子，这些例子可以使摩尔定律达到485的所有阶数。我们使用代数方法来解决此问题，这一问题可以追溯到G. Higman从关联方案理论（在有限几何中也称为Benson引理）的想法出发，表明不存在某些阶数的Moore Cayley有向图。