The Dilating Method for Cayley digraphs on finite Abelian groups

TL;DR

This paper introduces a geometric dilating method to generate infinite families of dense Cayley digraphs on finite Abelian groups, achieving maximum known density for degree 3.

Contribution

It presents a novel geometric approach for constructing dense Cayley digraphs with constant density across infinite families, applicable to any degree.

Findings

Method produces infinite families of dense Cayley digraphs

Achieves maximum known density for degree 3

Applicable to any degree for constant density

Abstract

A geometric method for obtaining an infinite family of Cayley digraphs of constant density on finite Abelian groups is presented. The method works for any given degree and it is based on consecutive dilates of a minimum distance diagram associated with a given initial Cayley digraph. The method is used to obtain infinite families of dense or asymptotically dense Cayley digraphs. In particular, for degree $d=3$, an infinite family of maximum known density is proposed.