  The jump graph $J(G)$ of a simple graph $G$ has vertices which represent
edges in $G$ where two vertices in $J(G)$ are adjacent if and only if the
corresponding edges in $G$ do not share an endpoint. In this paper, we examine
sequences of graphs generated by iterating the jump graph operation and
characterize the behavior of this sequence for all initial graphs. We build on
work by Chartrand et al. who showed that a handful of jump graph sequences
terminate and two sequences converge. We extend these results by showing that
there are no non-trivial repeating sequences of jump graphs. All diverging jump
graph sequences grow without bound while accumulating certain subgraphs.
