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 two or three. 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 6-valent first-kind Frobenius circulants. As a corollary 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. A distributed real-time computing architecture known as HARTS or hexagonal mesh is a special 6-valent first-kind Frobenius circulant.

Recursive cubes of rings as models for interconnection networks

“…In (30) we used the fact that, for a fixed x with 0 ≤ x ≤ q − 2, the number of elements (a, x) ∈ V is equal to 2 d(x+1) . Since 2 dq /(2 d − 1) ≤ 2 n , we have…”

Forwarding and optical indices of 4-regular circulant networks

“…Circulant graphs, or multi-loop networks as used in computer science literature, are basic structures for interconnection networks [5]. As such a lot of research on circulant graphs has been done in more than three decades, leading to a number of results on various aspects of circulant graphs [5,11,14,15,18,21,24,25,26,27]. Nevertheless, our knowledge on how circulant networks behave with regard to information dissemination is very limited.…”

“…In [23] it was proved that orbital regular graphs (which are essentially Frobenius graphs [10] except cycles and stars) achieve the smallest possible edge-forwarding index. In [25,26,27], Thomson and Zhou gave formulas for the edge-forwarding and arc-forwarding indices of two interesting families of Frobenius circulant graphs. The exact value of edge-forwarding index of some other graphs have also been computed, including Knödel graphs [11] and recursive circulant graphs [14].…”

Cyclotomic graphs and perfect codes

“…In the case when m = 3, 4, G 3 ((α)) and G 4 ((α)) are precisely the Eisenstein-Jacobi and Gaussian networks [13,24,26], respectively, where (α) is the principal ideal generated by 0 = α ∈ Z[ζ m ]. These two special families of cyclotomic graphs are closely related to two families of Frobenius circulants as shown in [41,Lemma 5] and [36,Theorem 5].…”

“…In general, it is challenging to construct perfect codes in Cayley graphs -many Cayley graphs do not contain any perfect code at all. Inspired by [24] and our own work [36,40] on Frobenius graphs, in this paper we study the following two families of Cayley graphs and perfect codes in them. Let ζ m (m ≥ 2) be an mth primitive root of unity, say ζ m = e 2πi/m , and A a nonzero ideal of the ring Z[ζ m ] of algebraic integers in the cyclotomic field Q(ζ m ).…”