Bartholdi Zeta Functions of Group Coverings of Digraphs

Graphs and digraphs treated here are finite and simple. Let G = (V (G), E(G)) be a connected graph with vertex set V (G) and edge set E(G), and D the symmetric digraph corresponding toG. SetD(G) = {(u, v), (v, u) | uv ∈ E(G)}. For e = (u, v) ∈ D(G), set u = o(e) and v = t(e). Furthermore, let e−1 = (v, u) be the inverse of e = (u, v). A path P of length n in G is a sequence P = (e1, · · · , en) of n arcs such that ei ∈ D(G), t(ei) = o(ei+1)(1 ≤ i ≤ n− 1). If ei = (vi−1, vi), 1 ≤ i ≤ n, then we also denote P = (v0, v1, · · · , vn). Set | P |= n, o(P ) = o(e1) and t(P ) = t(en). Also, P is called (o(P ), t(P ))-path. We say that a path P = (e1, · · · , en) has a backtracking if e−1 i+1 = ei for some i(1 ≤ i ≤ n − 1). A (v, w)-path is called a v-cycle (or v-closed path) if v = w. In graph theory, paths and cycles are diwalks and closed diwalks, respectively. We introduce an equivalence relation between cycles. Two cycles C1 = (e1, · · · , em) and C2 = (f1, · · · , fm) are called equivalent if fj = ej+k for all j. Let [C] be the equivalence class which contains a cycle C. Let B be the cycle obtained by going r times around a cycle B. Such a cycle is called a multiple of B. A cycle C is reduced if C has no backtracking. Furthermore, a cycle C is prime if it is not a multiple of a strictly smaller cycle. Note that each equivalence class of prime, reduced cycles of a graph G corresponds to a unique conjugacy class of the fundamental group π1(G, v) of G at a vertex v of G.