The largest order $n(d,k)$ of a graph of maximum degree $d$ and diameter $k$ cannot exceed the Moore bound, which has the form $M(d,k)=d^k - O(d^{k-1})$ for $d\to\infty$ and any fixed $k$. Known results in finite geometries on generalised $(k+1)$-gons imply, for $k=2,3,5$, the existence of an infinite sequence of values of $d$ such that $n(d,k)=d^k - o(d^k)$. This shows that for $k=2,3,5$ the Moore bound can be asymptotically approached in the sense that $n(d,k)/M(d,k)\to 1$ as $d\to\infty$; moreover, no such result is known for any other value of $k\ge 2$. The corresponding graphs are, however, far from vertex-transitive, and there appears to be no obvious way to extend them to vertex-transitive graphs giving the same type of asymptotic result. The second and the third author (2012) proved by a direct construction that the Moore bound for diameter $k=2$ can be asymptotically approached by Cayley graphs. Subsequently, the first and the third author (2015) showed that the same construct 查看全文>>