Algebraic Characterization of the SSC ${\mathrm{\Delta }}_s(G_{n,r}^1)$

Abstract

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