Asymptotic enumeration of vertex-transitive graphs of fixed valency

TL;DR

This paper investigates the asymptotic growth of the number of isomorphism classes of vertex-transitive graphs with fixed valency, specifically Cayley graphs, showing that their logarithm scales quadratically with the logarithm of the order.

Contribution

It provides the first asymptotic enumeration of vertex-transitive Cayley graphs of fixed valency, establishing precise growth rates for their count as the order increases.

Findings

Logarithm of the number of such graphs grows as d times (log n)^2

Established asymptotic bounds for general valency d

Derived stronger results specifically for valency 3

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 $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order 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 $d=3$.