Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$

Question

Prove, disprove, or give a counterexample:

Let $G$ be a graph with radius $r$ and let $x$ be a vertex in the center of $G$. If $d$($x,y$)=$r$, then $y$ is in the periphery of $G$.

Eccentricity - the greatest distance from v to any vertex

Radius - the value of the smallest eccentricity

Diameter - the value of the greatest eccentricity

Center - the set of vertices, v, such that the eccentricity is equal to the radius

Periphery - the set of vertices, u, such that the eccentricity equals the diameter.

Answer 1

Consider the kite ($K_4-e$).

The radius $r$ is 1. The diameter is 2.

The center is formed by the two vertices of degree 3, call them $x$ and $y$. The periphery is formed by the two other vertices.

Now $d(x,y)=r=1$, and $x$ is in the center, but $y$ is not in the periphery, it is even in the center.