If $G$ is a graph with distance function $d(x,y)$ between vertices, the **transmission** of a vertex $x \in v(G)$ is defined as $\sigma_{x}=\sum_{y \neq x}{d(x,y)}$. I want to know if there is a known characterization of graphs for which $\sigma_{x}$ is equal to the same number for all $x \in V(G)$.

Some easy examples: $K_{n},K_{p,p},C_{n}$. However, I found some non-regular examples as well.

UPDT: Those non-regular examples were a mistake in calculations - I still don't know if there are other, true ones.

UPDT2: As Gordon Royle pointed out, regular graphs of diameter 2 are transmission-unique; it can be easily shown that a transmission-unique graph of diameter 2 must be regular. He gave an example of diameter 3. Is there anything interesting to be said about such graphs of high diameter?