The center of a graph is the set of vertices with minimum eccentricity, so a $C_n$ have the minimum eccentricity equals to $n$ and a $K_n$ have the minimum eccentricity equals to $1$? Is valid that a center of a graph be the graph itself?
Let $G = (V, E)$ be a graph. If the eccentricity of each vertex $v \in V$ is $c$. Then the center of the graph is $V$, the set of all vertices of the graph $G$. Note that the center is the set $V$ of vertices not the graph $G$. In general the center of a graph $G = (V, E)$ is a subset of $V$.