Spanning Simplicial Ccomplexes of Uni-Cyclic Graphs

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.