Iterated Jump Graphs

TL;DR

This paper studies the behavior of sequences generated by repeatedly applying the jump graph operation to initial graphs, revealing that all diverging sequences grow unbounded and accumulate specific subgraphs.

Contribution

It extends previous work by characterizing the long-term behavior of all jump graph sequences, proving no non-trivial repeating sequences exist.

Findings

All diverging sequences grow without bound

Diverging sequences accumulate certain subgraphs

No non-trivial repeating jump graph sequences exist

Abstract

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.