On the periods of some graph transformations.

For any undirected graph with arbitrary integer values attached to the vertices, simultaneous updates are performed on these values, in which the value of a vertex is moved by 1 in the direction of the average of the values of the neighboring vertices. (A special rule applies when the value of a vertex equals this average.) It is shown that these transformations always reach a cycle of length 1 or 2. This proves a generalization of a conjecture made by Ghiglia and Mastin in connection with their work on a "phase-unwrapping" algorithm.