A Note on Moore Cayley Digraphs

Abstract

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: $$\begin{aligned} v\le (z+t)^2+z+1, \end{aligned}$$ 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.