Asymptotically approaching the Moore bound for diameter three by Cayley graphs

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 construction can be derived from generalised triangles with polarity. By a detailed analysis of regular orbits of suitable groups of automorphisms of graphs arising from polarity quotients of incidence graphs of generalised quadrangles with polarity, we prove that for an infinite set of values of $d$ there exist Cayley graphs of degree $d$, diameter $3$, and order $d^3{-}O(d^{2.5})$. The Moore bound for diameter $3$ can thus as well be asymptotically approached by Cayley graphs. We also show that this method does not extend to constructing Cayley graphs of diameter $5$ from generalised hexagons with polarity.