Generating Graphs with Symmetry

TL;DR

This paper introduces an algorithm for generating graphs with specific symmetries, enabling the study of symmetry effects on network dynamics and exploring the relationship between automorphism orbits and equitable partitions.

Contribution

The authors present a novel algorithm for creating graphs with prescribed symmetries and equitable partitions, addressing limitations of existing graph generation methods.

Findings

Generated graphs with desired symmetry properties.

Analyzed the relation between automorphism orbits and equitable partitions.

Provided tools for symmetry-based graph analysis.

Abstract

In the field of complex networks and graph theory, new results are typically tested on graphs generated by a variety of algorithms such as the Erd\H{o}s-R\'{e}nyi model or the Barab\'{a}si-Albert model. Unfortunately, most graph generating algorithms do not typically create graphs with symmetries, which have been shown to have an important role on the network dynamics. Here, we present an algorithm to generate graphs with prescribed symmetries. The algorithm can also be used to generate graphs with a prescribed equitable partition but possibly without any symmetry. We also use our graph generator to examine the recently raised question about the relation between the orbits of the automorphism group and a graph's minimal equitable partition.