# A new record of graph enumeration enabled by parallel processing

## TL;DR

This paper reports a breakthrough in graph enumeration using supercomputers, expanding known sequences and discovering optimal regular graphs for network design, by adapting classical algorithms for parallel processing.

## Contribution

It introduces a parallel processing method for enumerating regular graphs, enabling the computation of larger graph classes and discovering optimal graphs for network applications.

## Key findings

- Enumerated 4-regular graphs of order 23, setting a new record.
- Discovered several optimal regular graphs with minimal ASPL.
- Demonstrated the effectiveness of parallel adaptation of GENREG.

## 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 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 serval optimal regular graphs with minimum average shortest path lengths (ASPL) that can be used as interconnection networks for parallel computers. The number of 4-regular graphs and the optimal graphs, extremely time-consuming to calculate, result from a method we adapt from GENREG, a classical regular graph generator, to fit for supercomputers' strengths of using thousands of processor cores.