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

Abstract

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.

12 pages