Enumeration of Graph Coverings and related Surface Topology

G: a connected finite simple graph with vertex set V (G) and edge set E(G). Neighborhood N(v) of v : the set of vertices adjacent to v ∈ V (G). A map p : G̃ → G is a covering of G if p : V (G̃) → V (G) is surjective and p|N(ṽ) : N(ṽ) → N(v) is bijective for ∀v ∈ V (G) and ∀ṽ ∈ p−1(v). It is an n-fold covering if p is n-to-one. A covering p : G̃ → G is regular (or, A-covering) if there is a subgroup A of the automorphism group Aut (G̃) of G̃ acting freely on G̃ so that the graph G is isomorphic to the quotient graph G̃/A, say by h, and the quotient map G̃ → G̃/A is the composition h ◦ p of p and h. Betti number β(G) of G: the number of independent cycles in G. D(G): the set of all directed edges risen from the edges in E(G). Iso (G; n) (resp. IsoR(G; n)): the number of isomorphism classes of (resp. regular) n-fold coverings of G. Isoc (G; n) (resp. IsocR(G; n)): the number of isomorphism classes of connected (resp. regular) n-fold coverings of G. Iso (G;A) (resp. Isoc (G;A)): the number of isomorphism classes of (resp. connected) A-coverings of G. Sn, Zn and D n : the symmetric group on n letters, the cycle group of order n and the dihedral group of order 2n, respectively. Permutation voltage assignment of G: a function φ from D(G) into Sn with the property that φ(vu) = φ(uv)−1 for each uv ∈ D(G). Ordinary A-voltage assignment of G: a function φ from D(G) into a group A with the property that φ(vu) = φ(uv)−1 for each uv ∈ D(G). Permutation derived graph G : V (G) = V (G)× {1, 2, · · · , n}, and (u, i) and (v, j) are adjacent iff e = uv ∈ D(G) and j = φ(e)(i). The first coordinate projection p : G → G is an n-fold covering. Ordinary derived graph G ×φ A : V (G ×φ A) = V (G) ×A, and (u, g) and (v, h) are adjacent iff e = uv ∈ D(G) and h = φ(e)g. The first coordinate projection pφ : G ×φ A → G is an A-covering. Two coverings pi : G̃i → G, i = 1, 2, are said to be isomorphic if there exists a graph isomorphism 8 : G̃1 → G̃2 such that p2 ◦8 = p1. Such a 8 is called a covering isomorphism. 11991 Mathematics Subject Classification: 57M12, 57M15, 05C30.