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

Abstract

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 \textit{1-edge-connected chain} of \( r \) cycles \( C^1_r \) and \( G^1_{n,r} \), where \( C^1_r \) is a \textit{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 \( I_F(\Delta_s(G^1_{n,r})) \). Furthermore, we prove that the face ring \( k[\Delta_s(G^1_{n,r})] \) is Cohen-Macaulay.