Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.
In this paper, the ranks of a special family of Maiorana-McFarland bent functions are discussed. The upper and lower bounds of the ranks are given and those bent functions whose ranks achieve these bounds are determined. As a consequence, the inequivalence of some bent functions are derived. Furthermore, the ranks of the functions of this family are calculated when t 6.
In this paper we extend the theory of Grbner bases to difference-differential modules and present a new algorithmic approach for computing the Hilbert function of a finitely generated difference-differential module equipped with the natural filtration. We present and verify algorithms for construct-ing these Grbner bases counterparts. To this aim we introduce the concept of "generalized term order" on Nm ×Zn and on difference-differential modules. Using Grbner bases on difference-differential mod-ules we present a direct and algorithmic approach to computing the difference-differential dimension polynomials of a difference-differential module and of a system of linear partial difference-differential equations.
