Graph Coverings and Their Generalization

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We consider two objects in graph coverings and their generalization : regular coverings ; g-cyclic A-covers. We study three enumerations in this repot.The general problem of counting the ismorphism classes of regular n-fold coverings of a graph G with respect to a group GAMMA of automorphisms of G is still unsolved except in the case that n is prime. The enumeration of GAMMA-isomorphism classes of regular 4-fold coverings of G is a narural problem. A regular 4-fold covering of G is either a Z_2*Z_2-covering or a Z_4-covering of G.We enumerate the GAMMA-isomorphism classes of Z_2*Z_2-coverings of a connected graph.Next, for a connected symmetric digraph D,a finite group A and g<not a member of>A,we introduce a g-cyclic A-cover of D as a generalization of a regular covering of a graph. In the case that A is an abelian group with some property and the order of g is odd, we present a characterization for two g-cyclic A-covers of D to be ismorphic with respect to a group GAMMA of automorphisms of D.Thus, we enumerate the I-isomophism classes of g-cyclic Z_<pn>-covers of D for any p (>2). Furthermore, we count the GAMMA-isomophism classes of g-cyclic Z_p-covers of D.Related to connected g-cyclic a-covers, we present a decomposition formula for the number of I-isomorphism classes of g-cyclic A-covers of a connected symmetric digraph D for any finite abelian group A and any g<not a member of>A of odd order. Furthermore, we enumerate the I-isomorphism classes of g-cyclic A-covers of D,when A is the cyclic group Z_<pn> and the direct sum of m copies of Z_p for any prime number p (>2).