Large butterfly Cayley graphs and digraphs

TL;DR

This paper introduces new large Cayley graphs and digraphs inspired by butterfly networks, achieving record sizes for given diameter and degree, and approaching theoretical bounds.

Contribution

It presents novel constructions of Cayley graphs and digraphs with maximal order for specified diameter and degree, improving on previous bounds.

Findings

Largest known Cayley graphs and digraphs for certain parameters

Construction methods related to butterfly networks

Graphs approaching Moore bound in directed case

Abstract

We present families of large undirected and directed Cayley graphs whose construction is related to butterfly networks. One approach yields, for every large $k$ and for values of $d$ taken from a large interval, the largest known Cayley graphs and digraphs of diameter $k$ and degree $d$. Another method yields, for sufficiently large $k$ and infinitely many values of $d$, Cayley graphs and digraphs of diameter $k$ and degree $d$ whose order is exponentially larger in $k$ than any previously constructed. In the directed case, these are within a linear factor in $k$ of the Moore bound.