In this paper, we give a new and efficient algebraic criterion for the pure as well as non-pure shellability of simplicial complex ∆ 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 ∆ Γ (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.

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An algorithmic proof of the General Néron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case (cf.[10], [7]).

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them. An electrical network can be constructed from a graph by replacing each edge with a unit resistor. Resistance distances are computed by methods of the theory of resistive electrical networks (based on Ohm's and Kirchhoff's laws). The standard method to compute resistance distance is via the Moore-Penrose generalized inverse of the Laplacian matrix of the underlying graph G. In this article, we used the electric network approach and the combinatorial approach, to derive the exact expression for resistance distances between any two vertices of the Kn 4 ring model. By employing a recursive connection, firstly we determined all the eigenvalues and their multiplicities in relation to the corresponding Laplacian and normalised matrices. Secondly, we obtained the Kirchhoff index and multiplicative degree Kirchhoff index for Kn 4. Finally, we calculated the mean first passage time and Kemeny constant of Kn 4. Our computed results provide a comprehensive approach for exploring random walks on complex networks, especially biased random walks, which may also help to better understand and tackle some practical problems such as search and routing on networks.

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.