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A graph is called Frobenius if it is a connected orbital regular graph of a Frobenius group. A Frobenius map is a regular Cayley map whose underlying graph is Frobenius. In this paper, we show that almost all low-rank Frobenius graphs admit regular embeddings and enumerate non-isomorphic Frobenius maps for a given Frobenius graph. For some Frobenius groups, we classify all Frobenius maps derived from these groups. As a result, we construct some Frobenius maps with trivial exponent groups as a partial answer of a question raised by Nedela and Škoviera (Exponents of orientable maps, Proc. London Math. Soc. 75(3) (1997) 1–31).

The automorphism groups i>Aut(i>C(i>G, i>X)) and i>Aut(i>CM(i>G, i>X, i>p)) of a Cayley graph i>C(i>G, i>X) and a Cayley map i>CM(i>G, i>X, i>p) both contain an isomorphic copy of the underlying group i>G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group i>G by the stabilizer subgroup of the vertex 1i>G. We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.

This paper deals with numerical methods for the discretization of highly oscillatory systems. We approach the problem by writing the solution in terms of the Magnus expansion based on the Cayley map. The global error, obtained when the method is applied to the linear oscillator, is investigated. Moreover, we provide numerical experiments in order to validate our theoretical results.

In this paper we deal with high oscillatory systems and numerical methods for the approximation of their solutions. Some classical schemes developed in the literature are recalled and a recent approach based on the expression of the oscillatory solution by means of the exponential map is considered. Moreover we introduce a new method based on the Cayley map and provide some numerical tests in order to compare the different approaches.

A comparison is made, between three methods for visualizing the Minkowski space, in view of their application to relativistic kinematics.The velocity manifold method, in which a vector is represented by a point on the upper sheet of the unit-radius hyperboloid, presents two advantages: there is no distortion of angles, and the method is independent of the choice of any particular frame or observer. It is, however, limited to the representation of positive timelike vectors.The Cayley map method allows the representation of spacelike as well as timelike vectors, but it has two drawbacks: it distorts the angles, and it depends on the choice of a reference system.In the third method, the use of a unified space–time formalism permits one to visualize any kind of vector. This method generalizes the velocity manifold method, and presents the same two advantages: it provides us with a conformal representation, which is independent of any particular observer.We also present a rather concise comparison between two covariant methods of calculation, considered from the point of view of relativistic kinematics. One method is the familiar tensor calculus; the other is the spinor calculus, based on the representation of four-vectors by matrices.

A Cayley map M=CM(G,S,p) is t-balanced if p(x)^-^1=p^t(x^-^1) for every [email protected]?S. In this paper, we classify the regular t-balanced Cayley maps on semi-dihedral groups for any t. As a result, for every positive integer m, 4m-valent regular (2m+1)-balanced Cayley maps on semi-dihedral groups are constructed.

The purpose of this paper is to study self-dual embeddings of balanced Cayley maps. Given a Cayley map, necessary and sufficient conditions are given in terms of its underlying group for the map to be isomorphic to its dual embedding. Applications include self-dual embeddings of 2 n -dimensional cubes.