Relation between the center of two isomorphic graphs

Question

IF we have two isomorphic graphs $G_1$ and $G_2$ with:

$C(G_1) , C(G_2)$ : the Center of the graph $G_1$ and $G_2$ respectively, and they contain all vertices with minimum eccentricity $\varepsilon(v)$ by definition.

Can we say that for any two isomorphic graphs the following holds:

$$|C(G_1)| = |C(G_2)|$$

Thanks in advance.

Answer 1

Since relabelling the vertices does not affect the distances between them, the centres of two isomorphic graphs are the same (up to relabelling).