Algebraic Characterization of the SSC \(\Delta_s(G^1_{n,r})\)

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