  A multigraph is a nonsimple graph which is permitted to have multiple edges,
that is, edges that have the same end nodes. We introduce the concept of
spanning simplicial complexes $\Delta_s(\mathcal{G})$ of multigraphs
$\mathcal{G}$, which provides a generalization of spanning simplicial complexes
of associated simple graphs. We give first the characterization of all spanning
trees of a uni-cyclic multigraph $\mathcal{U}_{n,m}^r$ with $n$ edges including
$r$ multiple edges within and outside the cycle of length $m$. Then, we
determine the facet ideal $I_\mathcal{F}(\Delta_s(\mathcal{U}_{n,m}^r))$ of
spanning simplicial complex $\Delta_s(\mathcal{U}_{n,m}^r)$ and its primary
decomposition. The Euler characteristic is a well-known topological and
homotopic invariant to classify surfaces. Finally, we device a formula for
Euler characteristic of spanning simplicial complex
$\Delta_s(\mathcal{U}_{n,m}^r)$.
