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

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 $k[{\mathrm{\Delta }}_s({\mathcal{G}}_{n,r}^1)]$ for the spanning simplicial complex ${\mathrm{\Delta }}_s({\mathcal{G}}_{n,r}^1)$. Also, we characterize associated primes of the facet ideal $I_{$\mathcal{F}$} (\Delta_s ($\mathcal{G}${n,r}^1))$\mathrm{.}Furthermore,weprovethatthefacering$k[\Delta_s($\mathcal{G}${n,r}^{1})]$ is Cohen-Macaulay.