Cayley graphs of diameter two from difference sets

TL;DR

This paper constructs large diameter-2 Cayley graphs using difference sets, improving lower bounds on their maximum order for certain degrees, especially in abelian groups.

Contribution

It introduces new constructions of large Cayley graphs of diameter two based on generalized difference sets, enhancing known lower bounds.

Findings

Established a lower bound of (25/64)d^2 - 2.1 d^{1.525} for large d.

Proved a lower bound of (4/9)d^2 for degrees d=3q with q=2^m and m odd.

Compared new bounds with existing ones, showing significant improvements.

Abstract

Let $C(d,k)$ and $AC(d,k)$ be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree $d$ and diameter $k$. When $k=2$, it is well-known that $C(d,2)\le d^2+1$ with equality if and only if the graph is a Moore graph. In the abelian case, we have $AC(d,2)\le \frac{d^2}{2}+d+1$. The best currently lower bound on $AC(d,2)$ is $\frac{3}{8}d^2-1.45 d^{1.525}$ for all sufficiently large $d$. In this paper, we consider the construction of large graphs of diameter $2$ using generalized difference sets. We show that $AC(d,2)\ge \frac{25}{64}d^2-2.1 d^{1.525}$ for sufficiently large $d$ and $AC(d,2) \ge \frac{4}{9}d^2$ if $d=3q$, $q=2^m$ and $m$ is odd.