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.

A first kind Frobenius graph is a Cayley graph Cay(K , S) on the Frobenius kernel of a Frobenius group K H such that S = a H for some a ∈ K with a H = K , where H is of even order or a is an involution. It is known that such graphs admit 'perfect' routing and gossiping schemes. A circulant graph is a Cayley graph on a cyclic group of order at least three. Since circulant graphs are widely used as models for interconnection networks, it is thus highly desirable to characterize those which are Frobenius of the first kind. In this paper we first give such a characterization for connected 4-valent circulant graphs, and then describe optimal routing and gossiping schemes for those which are first kind Frobenius graphs. Examples of such graphs include the 4-valent circulant graph with a given diameter and maximum possible order.2000 Mathematics subject classification: 05C25, 68M10, 90B18.

Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) triple-loop network TL n (a, b, c) is the degree-six graph with vertices 0, 1, 2, . . . , n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the operation is modulo n. It is known that the maximum order of a connected triple-loop network of the form TL n (a, b, n − (a + b)) with given diameter d ≥ 2 is n d = 3d 2 + 3d + 1, which is achieved by2 −1). In this article, we study the routing and gossiping problems for such optimal tripleloop networks under the store-and-forward, all-port, and full-duplex model, and prove that they admit "perfect" gossiping and routing schemes which exhibit many interesting features. Using a group-theoretic approach we develop for TL n d a method for systematically producing such optimal gossiping and routing schemes. Moreover, we determine the minimum gossip time, the edge-and arc-forwarding indices, and the minimal edge-and arcforwarding indices of TL n d , and prove that our routing schemes are optimal with respect to these four indices simultaneously. As a key step towards these results, we prove that TL n d is a Frobenius graph on a Frobenius group with Frobenius kernel Z n d , and that TL n d is arc-transitive with respect to this Frobenius group. In addition, we show that TL n d admits complete rotations.

A Frobenius group is a transitive permutation group which is not regular but only the identity element can fix two points. Such a group can be expressed as the semi-direct product G = K ⋊ H of a nilpotent normal subgroup K and another group H fixing a point. A firstkind G-Frobenius graph is a connected Cayley graph on K with connection set an H-orbit a H on K that generates K, where H has an even order or a is an involution. It is known that the first-kind Frobenius graphs admit attractive routing and gossiping algorithms. A complete rotation in a Cayley graph on a group G with connection set S is an automorphism of G fixing S setwise and permuting the elements of S cyclically. It is known that if the fixed-point set of such a complete rotation is an independent set and not a vertex-cut, then the gossiping time of the Cayley graph (under a certain model) attains the smallest possible value. In this paper we classify all first-kind Frobenius circulant graphs that admit complete rotations, and describe a means to construct them. This result can be stated as a necessary and sufficient condition for a first-kind Frobenius circulant to be 2-cell embeddable on a closed orientable surface as a balanced regular Cayley map. We construct a family of non-Frobenius circulants admitting complete rotations such that the corresponding fixed-point sets are independent and not vertex-cuts. We also give an infinite family of counterexamples to the conjecture that the fixed-point set of every complete rotation of a Cayley graph is not a vertex-cut.

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