Abelian Cayley digraphs with asymptotically large order for any given degree

TL;DR

This paper introduces a new family of Abelian Cayley digraphs constructed via a generalized congruence approach, achieving asymptotically large order for any fixed degree as diameter increases, providing explicit constructions where previous results were non-constructive.

Contribution

It presents a constructive method to generate Abelian Cayley digraphs with asymptotically large order for fixed degree, improving upon prior non-constructive results.

Findings

Constructed a family of Abelian Cayley digraphs with large order

Achieved asymptotic growth of vertices with increasing diameter

Provided explicit constructions surpassing previous non-constructive results

Abstract

Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$ of the concept of congruence in $Z$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known asymptotically dense results were all non-constructive.