Asymptotic enumeration of vertex-transitive graphs of fixed valency

Abstract

Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating 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 $u{v}^{-1}\in S$. Let $CA{Y}_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log (CA{Y}_d(n))\in \Theta (d(\log n{)}^2)$, as $n\to \infty$. We also obtain some stronger results in the case $d=3$.

15 pages

References in corpus (1)

• Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs

Cited by in corpus (1)

• Groups of order at most 6 000 generated by two elements, one of which is an involution, and related structures