A family of efficient six-regular circulants representable as a Kronecker product

Broere and Hattingh proved that the Kronecker product of two circulants whose orders are co-prime is a circulant itself. This paper builds on this result to construct a family of efficient three-colorable, six-regular circulants representable as the Kronecker product of a Möbius ladder and an odd cycle. The order of each graph is equal to 4d2−2d−24d2−2d−2 where dd denotes the diameter and d≡3,5d≡3,5 (mod 6). Additional results include (a) distance-wise vertex distribution of the circulant leading to its average distance that is about two-thirds of the diameter, (b) routing via shortest paths, and (c) an embedding of the circulant on a torus with a half twist. In terms of the order–diameter ratio and odd girth, the circulants in this paper surpass the well-known triple-loop networks having diameter dd and order 3d2+3d+13d2+3d+1.