to check the required property of a given graph. Can a graph be self-centered if it contains a vertex of degree one. The simplest counter example that came to my mind is Path.
But how to prove the statement if we consider any graph with a vertex of degree one.
A self-centered graph is a graph whose diameter equals its radius. Or where the eccentricity of every vertex is the same.

Let $a$ be a vertex of degree one, $b$ a vertex adjacent to $a$. Then ${\rm eccentricity}(a)={\rm eccentricity}(b)+1$ if the graph contains at least 3 vertices.
