Cayley graphs of diameter two with order greater than 0.684 of the Moore bound for any degree

TL;DR

This paper establishes a new lower bound for the order of Cayley graphs with diameter two, showing they can have more vertices than previously proven, for all degrees above a large threshold.

Contribution

It provides a novel lower bound of 0.684d^2 for Cayley graphs of diameter two, improving upon existing bounds for sufficiently large degrees.

Findings

Lower bound C(d,2)>0.684d^2 for degrees d≥360756

Significant improvement over previous bounds

Applicable to all degrees above a large threshold

Abstract

It is known that the number of vertices of a graph of diameter two cannot exceed $d^2+1$. In this contribution we give a new lower bound for orders of Cayley graphs of diameter two in the form $C(d,2)>0.684d^2$ valid for all degrees $d\geq 360756$. The result is a significant improvement of currently known results on the orders of Cayley graphs of diameter two.