The permutation cycle graph is defined as follows.
Permutation Cycle Graph: A permutation cycle graph for a given permutation $\pi$ of a finite set $V$, is its cycle graph $\Gamma$ such that the vertex set is $V$ and the edges are pairs $\left(\mu, \omega\right)$ for which $\pi(\mu)=\omega$.
In that case for any $\pi$ with more than one cycle, the graph is the disjoint union of at least two cycles. So, the eccentricity any vertex is infinite.
Am I getting it right?