The **imbalance** of an edge $(u,v) \in E(G)$ of a graph $G$ is defined as $|d(u)-d(v)|$ ($d$ being, as usual the degree). (This concept was introduced by Albertson in 1997)

I'm interested in the set of possible imbalances in a graph. In particular I am looking for examples of graphs where all edges have the same imbalance. Regular graphs trivially have this property and so does the path $P_{3}$.

Is it true that if all edges of $G$ have the same imbalance, then $G$ is either regular or $P_{3}$?

I have an inkling that this ought to be true by some sort of pigeonhole principle but can't think up a proof. A counterexample will be even more welcome :)