Large Networks of Diameter Two Based on Cayley Graphs

TL;DR

This paper constructs large, diameter-two networks using Cayley graphs with simple groups, achieving near-optimal size for various degrees and identifying the largest known vertex-transitive graphs of diameter two for specific degrees.

Contribution

It introduces a new construction method for large diameter-two networks based on Cayley graphs with simple groups, improving known bounds for certain degrees.

Findings

Constructed Cayley graphs of diameter two with order > 2/3 Moore bound for small degrees

Presented a general construction for networks of order 0.5d^2 for d ≥ 8

Identified the largest known vertex-transitive graphs of diameter two for degrees 16-35

Abstract

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.