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 multigraph $G$, $\Delta_s(G)$ 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 $I_c(G)=(x_{i_1}\cdots x_{i_k}|\ \{x_{i_1},\ldots, x_{i_k}\}\ \textrm{is the edge set of a cycle in}\ G)$. Also, some homological invariants of the Stanley-Reisner ring of $\Delta_s(G)$ such as projective dimension and regularity are investigated.
