Enumeration of regular graph coverings having finite abelian covering transformation groups

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. An enumeration of the isomorphism classes of n-fold coverings of a graph G was done by Kwak and Lee [Canad. J. Math., XLII (1990), pp. 747-761] and independently by Hofmeister [Discrete Math., 98 (1991), pp. 437-444]. An enumeration of the isomorphism classes of connected n-fold coverings of a graph G was recently done by Kwak and Lee [J. Graph Theory, 23 (1996), pp. 105-109]. But the enumeration of the isomorphism classes of regular coverings of a graph G has been done for only a few cases. In fact, the isomorphism classes of A-coverings of G were enumerated when A is the cyclic group Z(n), the dihedral group D-n (n: odd), and the direct sum of m copies of Z(p). (See [Discrete Math., 143 (1995), pp. 87-97], [J. Graph Theory, 15 (1993), pp. 621-627], and [Discrete Math., 148 (1996), pp. 85-105]). In this paper, we discuss a method to enumerate the isomorphism classes of connected A-coverings of a graph G for any finite group A and derive some formulas for enumerating the isomorphism classes of regular n-fold coverings for any natural number n. In particular, we calculate the number of the isomorphism classes of A-coverings of G when A is a finite abelian group or the dihedral group D-n. Our method gives partial answers to the open problems 1 and 2 in [Discrete Math., 148 (1996), pp. 85-105] and also gives a formula to calculate the number of the subgroups of a given index of any finitely generated free abelian group.