In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph. Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space), and finite cyclic groups ℤn for n > 2. When is a cyclic group, the voltage graph may be called a cyclic-voltage graph. Definition Formal definition of a -voltage graph, for a given group : Begin with a digraph G. (The direction is solely for convenience in notation.) A -voltage on an arc of G is a label of the arc by an element x of . For instance, in the case where , the label is a number i (mod n). A -voltage assignment is a function that labels each arc of G with a Π-voltage. A -voltage graph is a pair such that G is a digraph and α is a voltage assignment. The voltage group of a voltage graph is the group from which the voltages are assigned. Note that the voltages of a voltage graph need not satisfy Kirchhoff's voltage law, that the sum of voltages around a closed path is 0 (the identity element of the group), although this law does hold for the derived graphs described below. Thus, the name may be somewhat misleading. It results from the origin of voltage graphs as dual to the current graphs of topological graph theory. The derived graph The derived graph of a voltage graph is the graph whose vertex set is and whose edge set is , where the endpoints of an edge (e, k) such that e has tail v and head w are and . Although voltage graphs are defined for digraphs, they may be extended to undirected graphs by replacing each undirected edge by a pair of oppositely ordered directed edges and by requiring that these edges have labels that are inverse to each other in the group structure. In this case, the derived graph will also have