Algebraic characterization of the SSC $Δ_s(\mathcal{G}_{n,r}^{1})$

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.