Algebraic properties of spanning simplicial complexes

Abstract

In this paper, we study some algebraic properties of the spanning simplicial complex $\Delta _s(G)$ associated to a multigraph~$G$. It is proved that, for any finite multi\-graph~$G$, $\Delta _s(G)$ is a pure vertex decomposable simplicial complex and therefore shellable and Cohen-Macaulay. As a consequence, we deduce that, for any multigraph~$G$, the quotient ring $R/I_c(G)$ is Cohen-Macaulay, where \[ I_c(G)=(x_{i_1} \cdots x_{i_k} \mid \{x_{i_1},\ldots , x_{i_k}\}\qquad \qquad \qquad \] \[ \qquad \qquad \qquad \mbox {is the edge set of a cycle in~$G$}). \] Also, some homological invariants of the Stanley-Reisner ring of $\Delta _s(G)$, such as projective dimension and regularity, are investigated.