  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.
