the property that its directed edges form pairs of oppositely oriented edges, so the derived graph may itself be interpreted as being an undirected graph. The derived graph is a covering graph of the given voltage graph. If no edge label of the voltage graph is the identity element, then the group elements associated with the vertices of the derived graph provide a coloring of the derived graph with a number of colors equal to the group order. An important special case is the bipartite double cover, the derived graph of a voltage graph in which all edges are labeled with the non-identity element of a two-element group. Because the order of the group is two, the derived graph in this case is guaranteed to be bipartite. Polynomial time algorithms are known for determining whether the derived graph of a -voltage graph contains any directed cycles. Examples Any Cayley graph of a group , with a given set of generators, may be defined as the derived graph for a -voltage graph having one vertex and self-loops, each labeled with one of the generators in . The Petersen graph is the derived graph for a ℤ5-voltage graph in the shape of a dumbbell with two vertices and three edges: one edge connecting the two vertices, and one self-loop on each vertex. One self-loop is labeled with 1, the other with 2, and the edge connecting the two vertices is labeled 0. More generally, the same construction allows any generalized Petersen graph GP(n,k) to be constructed as a derived graph of the same dumbbell graph with labels 1, 0, and k in the group ℤn. The vertices and edges of any periodic tessellation of the plane may be formed as the derived graph of a finite graph, with voltages in ℤ2. Notes References . . . . . . Extensions and generalizations of graphs