An Efficient Algebraic Criterion for Shellability

In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex $\mathrm{\Delta }$ over [n]. We also give an algebraic characterization of a leaf in a simplicial complex (defined in [8]). Moreover, we introduce the concept of Gallai-simplicial complex ${\mathrm{\Delta }}_{\mathrm{\Gamma }}(G)$ of a finite simple graph G. As an application, we show that the face ring of the Gallai simplicial complex associated to tree is Cohen-Macaulay.