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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 ongeneralised $(k+1)$-gons imply, for $k=2,3,5$, the existence of an infinitesequence 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 anyother value of $k\ge 2$. The corresponding graphs are, however, far fromvertex-transitive, and there appears to be no obvious way to extend them tovertex-transitive graphs giving the same type of asymptotic result.  The second and the third author (2012) proved by a direct construction thatthe Moore bound for diameter $k=2$ can be asymptotically approached by Cayleygraphs. Subsequently, the first and the third author (2015) showed that thesame construction can be derived from generalised triangles with polarity.  By a detailed analysis of regular orbits of suitable groups of automorphismsof graphs arising from polarity quotients of incidence graphs of generalisedquadrangles 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 beasymptotically approached by Cayley graphs. We also show that this method doesnot extend to constructing Cayley graphs of diameter $5$ from generalisedhexagons with polarity.

Asymptotically approaching the Moore bound for diameter three by Cayley graphs