Large circulant graphs of fixed diameter and arbitrary degree

TL;DR

This paper introduces new construction techniques for large circulant graphs with fixed diameter and arbitrary degree, improving bounds and providing applications in additive combinatorics.

Contribution

It presents a direct product construction and a novel stitching method to enhance bounds on circulant graphs of any diameter.

Findings

Improved bounds for small diameters using direct product construction.

Enhanced asymptotic orders for all diameters via stitching technique.

Applications to bounds on sumsets in cyclic groups.

Abstract

We consider the degree-diameter problem for undirected and directed circulant graphs. To date, attempts to generate families of large circulant graphs of arbitrary degree for a given diameter have concentrated mainly on the diameter 2 case. We present a direct product construction yielding improved bounds for small diameters and introduce a new general technique for "stitching" together circulant graphs which enables us to improve the current best known asymptotic orders for every diameter. As an application, we use our constructions in the directed case to obtain upper bounds on the minimum size of a subset $A$ of a cyclic group of order $n$ such that the $k$-fold sumset $kA$ is equal to the whole group. We also present a revised table of largest known circulant graphs of small degree and diameter.