A Structured Table of Graphs with Symmetries and Other Special Properties

Abstract

We organize a table of regular graphs with minimal diameters and minimal mean path lengths, large bisection widths and high degrees of symmetries, obtained by enumerations on supercomputers. These optimal graphs, many of which are newly discovered, may find wide applications, for example, in design of network topologies.

Optimal circulant graphs as low-latency network topologies

“…Deng et al implemented parallel exhaustive search of regular graphs [47] and random search with rotational symmetry to obtain small and large (near) optimal topologies and evaluated the performance by benchmarking both on real clusters and simulation platform [22]. More small optimal topologies with symmetries and other special properties were produced and presented in a structured table [53]. Distributed shortcut networks targeting the diameter and cable length trade-off [46] and host-switch graphs designed by minimizing diameter and/or MPL using randomized heuristics [48] were also introduced and benchmarked by simulation.…”

“…Recent research on constructing supercomputer network utilizing Lie algebra symmetry [21,41] and optimizing MPL by random or heuristic algorithms with rotational symmetry constraints [22,39] have demonstrated the importance of the symmetry perspective in network design. The total number of symmetries of a graph calculated as its automorphism group size has also been included in further optimization of smaller topologies [53]. Vertex-symmetric topologies are one class of highly symmetric topologies, where any two vertices are equivalent under self mappings that preserve adjacency [20].…”