  An elastic graph is a graph with an elasticity associated to each edge. It
may be viewed as a network made out of ideal rubber bands. If the rubber bands
are stretched on a target space there is an elastic energy. We characterize
when a homotopy class of maps from one elastic graph to another is loosening,
i.e., decreases this elastic energy for all possible targets. This fits into a
more general framework of energies for maps between graphs.
