I am trying to work out a problem. Given a self-centered graph, is the square of the graph also a self-centered graph? I tried numerically on few graphs given in http://www.matf.bg.ac.rs/~zstanic/indexdiam.html and I got the result that, yes, they are still self-centered. But I am not getting any idea to prove it theoretically.
A self-centered graph is a graph whose diameter equals its radius. Or where the eccentricity of every vertex is the same.
The $k$th power of a graph $G$ is a graph with the same set of vertices as $G$ and an edge between two vertices iff there is a path of length at most $k$ between them.