Asymptotic enumeration of vertex-transitive graphs of fixed valency

Let G be a group and let S be an inverse-closed and identity-free generating set of G. The Cayley graph Cay(G,S) has vertex-set G and two vertices u and v are adjacent if and only if uv−1∈S. Let CAYd(n) be the number of isomorphism classes of d-valent Cayley graphs of order at most n. We show that log⁡(CAYd(n))∈Θ(d(log⁡n)2), as n→∞. We also obtain some stronger results in the case d=3.