On the equivalence between a conjecture of Babai-Godsil and a conjecture of Xu concerning the enumeration of Cayley graphs

TL;DR

This paper proves that two longstanding conjectures about the asymptotic enumeration of Cayley graphs are actually equivalent, unifying two different perspectives in algebraic graph theory.

Contribution

It establishes the equivalence between Babai-Godsil's and Xu's conjectures, providing a broader theorem on Cayley graph enumeration.

Findings

The two conjectures are mathematically equivalent.

A general theorem on Cayley graph enumeration is presented.

Implications for future research in algebraic graph theory.

Abstract

In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.