  In this paper, we give a new algebraic criterion for the {\em shellability}
of (non-pure) simplicial complex $\Delta$ over $[n]$, shellable in the sense of
Bj\"orner and Wachs \cite{BW}. We show that the spanning simplicial complex of
doubly uni-cyclic graph is non-pure shellable. Moreover, we introduce the
concept of Gallai-simplicial complex $\Delta_{\Gamma}(G)$ of a finite simple
graph $G$. We applied the obtained criterion to discuss the shellability of
Gallai simplicial complexes associated to various classes of graphs..
