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Let $G$ be a group and let $S$ be an inverse-closed and identity-freegenerating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let$CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs oforder at most $n$. We show that $\log(CAY_d(n))\in\Theta (d(\log n)^2)$, as$n\to\infty$. We also obtain some stronger results in the case $d=3$.

Asymptotic enumeration of vertex-transitive graphs of fixed valency