Efficient eight-regular circulants based on the Kronecker product

This paper presents a family of efficient eight-regular circulants representable as the Kronecker product of the dense four-regular circulant on 2r2+2r+1 nodes and the cycle C4r+3, where r≡0,1,2,4 (mod 5). Each graph is of order 12(2d+1)(d2+1), where d denotes its diameter, d is odd, and d≡0,1,3,4 (mod 5). Its average distance is about two-thirds of its diameter. Other salient characteristics include high odd girth, three-colorability, and an edge decomposition into Hamiltonian cycles. The baseline of the present study is a theorem by Broere and Hattingh, which states that the Kronecker product of two circulants whose orders are co-prime is a circulant itself.