A New Record of Graph Enumeration Enabled by Parallel Processing

Zhipeng Xu,Fabian Jimenez,Yuefan Deng,Xiaolong Huang

doi:10.3390/math7121214

Abstract

Using three supercomputers, we broke a record set in 2011, in the enumeration of non-isomorphic regular graphs by expanding the sequence of A006820 in the Online Encyclopedia of Integer Sequences (OEIS), to achieve the number for 4-regular graphs of order 23 as 429,668,180,677,439, while discovering several regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The enumeration of 4-regular graphs and the discovery of minimal-ASPL graphs are extremely time consuming. We accomplish them by adapting GENREG, a classical regular graph generator, to three supercomputers with thousands of processor cores.

Highlights

The analysis of regular graphs for their properties, including eigen-spectra and automorphisms, is a fertile field for discovery and applications in algebraic graph theory [1,2,3]
We extend GENREG for distributed clusters by using the message passing interface (MPI) [23]
Using the parallel GENREG we developed, we obtained the following results: (1)

Summary

Introduction

The analysis of regular graphs for their properties, including eigen-spectra and automorphisms, is a fertile field for discovery and applications in algebraic graph theory [1,2,3]. For 3-regular graphs of order n, Robinson and Wormald [12,13] presented all counting results for n ≤ 40, while pointing out that enumeration for unlabeled k-regular graphs with k > 3 is an unsolved problem. Using the parallel GENREG we developed, we obtained the following results:. Kimberley [26] used GENREG to enumerate the 4-regular graphs for up to the order 22 in 2011 [27]. This record for n = 22 remained unchallenged until our enumeration for n = 23, enabled by our parallel computing implementation to advance it a step

The Enumeration Function

Search for a Regular Graph with Minimal ASPL

Conclusions