Connected graph may have precisely two non-adjacent central vertices

Statement

It is possible to construct a finite connected undirected graph that has precisely two central vertices that are not adjacent to each other.

Proof

Consider the graph with vertex set $\{1,2,3,4,5,6\}$ and edge set:

$\{\{1,2\},\{2,3\},\{2,4\},\{3,5\},\{4,5\},\{5,6\}\}$

We note below the eccentricities of the vertices:

Vertex	Farthest vertices from it	Eccentricity
1	6	4
2	6	3
3	1,6	2
4	1,6	2
5	1	3
6	1	4

We see from the above that the radius of the graph is 2 and that there are precisely two central vertices: vertex 3 and vertex 4. Moreover, we see that they are non-adjacent vertices.