On existing optimal three-dimensional circulant networks

The optimal circulant graphs have minimum diameter for the given order N and degree ν and, respectively, optimal features with respect to communication delays, reliability and connectivity under implementation as interconnection networks in multimodule supercomputer systems. The questions of existence of optimal (nearly optimal) circulants with the degree ν =  6 are investigated. An algorithm for enumeration of graphs, the experimental results obtained for optimal and nearly optimal loop circulant graphs of degree 6 and some conclusions about the probability of their existence are presented.