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.

Author Bio

Fran Herr graduated from the University of Washington in 2022 with degrees in mathematics and dance. She is a recipient of the 2022-2023 Exponent Fellowship from the National Museum of Mathematics. Her mathematical interests lie in graph theory, topology, complex analysis, and other visually driven subjects. Fran plans on attending graduate school to continue studying mathematics.

Legrand Jones II is a graduate student in mathematics at Indiana University Bloomington. He received his Bachelor of Science degrees in pure mathematics and physics in 2021 from the University of Washington.