  Let $G_n$ be a graph obtained by the strong product of $P_2$ and $C_n$, where
$n\geqslant3$. In this paper, explicit expressions for the Kirchhoff index,
multiplicative degree-Kirchhoff index and number of spanning trees of $G_n$ are
determined, respectively. It is surprising to find that the Kirchhoff (resp.
multiplicative degree-Kirchhoff) index of $G_n$ is almost one-sixth of its
Wiener (resp. Gutman) index. Moreover, let $\mathcal{G}^r_n$ be the set of
subgraphs obtained from $G_n$ by deleting any $r$ vertical edges of $G_n$,
where $0\leqslant r\leqslant n$. Explicit formulas for the Kirchhoff index and
the number of spanning trees for any graph $G^r_n\in \mathcal{G}^r_{n}$ are
completely established, respectively. Finally, it is interesting to see that
the Kirchhoff index of $G^r_n$ is almost one-sixth of its Wiener index.
