  Graph maps that are homotopic to the identity and that permute the vertices
are studied. Given a periodic point for such a map, a {\em rotation element} is
defined in terms of the fundamental group. A number of results are proved about
the rotation elements associated to periodic points in a given edge of the
graph. Most of the results show that the existence of two periodic points with
certain rotation elements will imply an infinite family of other periodic
points with related rotation elements. These results for periodic points can be
considered as generalizations of the rotation interval for degree one maps of
the circle.
