  In this paper, we characterize the set of spanning trees of
$\mathcal{G}_{n,r}^1$ (a simple connected graph consisting of $n$ edges,
containing exactly one $1$-edge-connected chain of $r$ cycles $\mathbb{C}_r^1$
and $\mathcal{G}_{n,r}^{1}\setminus\mathbb{C}_r^1$ is a forest). We compute the
Hilbert series of the face ring $k[\Delta_s (\mathcal{G}_{n,r}^1)]$ for the
spanning simplicial complex $\Delta_s (\mathcal{G}_{n,r}^1)$. Also, we
characterize associated primes of the facet ideal $I_{\mathcal{F}} (\Delta_s
(\mathcal{G}_{n,r}^1))$. Furthermore, we prove that the face ring
$k[\Delta_s(\mathcal{G}_{n,r}^{1})]$ is Cohen-Macaulay.
