On The Algebraic Study of Spanning Simplicial Complexes of \(r\)-cyclic Graphs \(G_{n,r}\)

Abstract

In this paper, we characterize all spanning trees of the \(r\)-cyclic graph \(G_{n,r}\). We provide the formulation of \(f\)-vectors associated with spanning simplicial complexes \(\Delta_s(G_{n,r})\) and, consequently, deduce a formula for computing the Hilbert series of the Stanley-Reisner ring \(k[\Delta_s(G_{n,r})]\). For the facet ideal \(I(\Delta(G_{n,r}))\), we characterize all associated primes. Specifically, for uni-cyclic graphs with cycle length \(m_i\), we prove that the facet ideal \(I(\Delta(G_{n,1}))\) has linear quotients with respect to its generating set. Furthermore, we establish that projdim \((I_{\mathcal{F}}(\Delta_s(G_{n,1}))) = 1\) and \(\beta_i(I_{\mathcal{F}}(\Delta_(G_s{n,1}))) = m_i\) for \(i \leq 1\).