You need an additional constraint/premise of some kind. It's not true as stated.
Let (eccentricity of v) and , where V(G) is the set of vertices of G and d(u,v) is the distance between u and v. Let (periphery of G) Pe(G) be the graph induced by the vertices of G that have eccentricity equivalent to diam(G). Now suppose that the graph H has n vertices. Prove that H is Pe(G) if and only if . ( is the maximum degree in H)