Diameters of random circulant graphs

TL;DR

This paper investigates the diameter growth of random circulant graphs, showing they scale as n^{1/k} and establishing a limit theorem for their diameter distribution, bridging small-world and linear growth behaviors.

Contribution

It provides the first rigorous analysis of diameter scaling and distribution for random circulant graphs, revealing their intermediate growth pattern.

Findings

Diameter scales as n^{1/k} for k-regular circulant graphs

Limit theorem for diameter distribution established

Results extend to average distance and higher moments

Abstract

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n^{1/k}, and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.