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

Abstract

A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as G = K ⋊ H with K a nilpotent normal subgroup. A first-kind G-Frobenius graph is a Cayley graph on K with connection set S an H-orbit on K generating K, where H is of even order or S consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel K is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its forwarding indices, Wiener indices and minimum gossip time. We also prove that its broadcasting time is equal to its diameter plus..