Diameter formula for generalized Petersen graphs

Let $GPG(n,s)$ be the generalized Petersen graph and let $C_n(1,s)$ be the circulant graph with step sizes $1$ and $s$. Let $n\geq 5$ and $2\leq s\leq \lfloor \frac{n-1}{2} \rfloor$. Diameter formula. We have

$$D(GPG(n,s)) = \begin{cases} D(C_n(1,s)) + 1 & \text{if } n=4p \text{ and } s=2p-1 \text{ with } p>2,\\ D(C_n(1,s)) + 2 & \text{otherwise.} \end{cases}$$

This gives an exact relation between the diameters of generalized Petersen graphs and the corresponding circulant graphs; together with the preceding diameter bounds for $C_n(1,s)$, it yields explicit upper bounds for $GPG(n,s)$.