I am studying on the graphs where eccentricity of every vertex is same. If $G$ is such graph where eccentricity is $r$ for every vertex and for a vertex $x$ if there exists at least two vertices such that $d(x,y) = d(x,z) =r$, then how to find number of vertices in the graph.

*My attempt:*

I considered two shortest paths $P$ and $P'$ between $x$ and $y$, and $x$ and $z$, respectively, such that lengths of these paths is $r$. So, total number of vertices traversed is $2r+2$ but the vertex $x$ is common, so finally I came to the conclusion that the graph has at least 2r+1 vertices, i.e., $n\geq 2r+1$.

Is my procedure true? Am I missing any important fact? Kindly help. Any hint or suggestion is welcome.. Thanks for the help.

What if some vertices are repeated. I am not getting any idea. Kindly help.