  In this paper, we introduce the concept of spanning simplicial complexes
$\Delta_s(G)$ associated to a simple finite connected graph G. We give the
characterization of all spanning trees of the uni-cyclic graph $U_{n,m}$. In
particular, we give the formula for computing the Hilbert series and h-vector
of the Stanley-Riesner ring k[\Delta_s(U_{n,m})]. Finally, we prove that the
spanning simplicial complex $\Delta_s(U_{n,m})$ is shifted hence
$\Delta_s(U_{n,m})$ is shellable.
