Frobenius circulant graphs of valency six, Eisenstein-Jacobi networks, and hexagonal meshes

A finite Frobenius group is a permutation group which is transitive but not regular such that only the identity element can fix two points. Such a group can be expressed as a semidirect product G = K o H, where K is a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K whose connection set is an H-orbit S on K that generates K, where H is of even order or S consists of involutions. In this paper we classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping, routing and broadcasting algorithms for such circulants and compute their forwarding indices, Wiener indices and minimum gossip time. We also prove that the broadcasting time of such a circulant is equal to its diameter plus two or three, indicating that it is efficient for broadcasting. We prove that all 6-valent first-kind Frobenius circulants with cyclic kernels are Eisenstein-Jacobi graphs, the latter being Cayley graphs on quotient rings of the ring of Eisenstein-Jacobi integers. We also prove that larger Eisenstein-Jacobi graphs can be constructed from smaller ones as topological covers, and a similar result holds for the family of 6-valent first-kind Frobenius circulants. As a corollary we show that any Eisenstein-Jacobi graph with order congruent to 1 modulo 6 and underlying Eisenstein-Jacobi integer not an associate of a real integer, is a cover of a 6-valent first-kind Frobenius circulant. We notice that a distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.