We introduce a class of simplicial complexes which we call Buchsbaum* over a field. Buchsbaum* complexes generalize triangulations of orientable homology manifolds. By definition, the Buchsbaum* property depends only on the geometric realization and the field. Characterizations in terms of simplicial and local cohomology are given. It is shown that Buchsbaum* complexes are doubly Buchsbaum. App...

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...

The Laplacian of an undirected graph is a square matrix, whose eigenvalues yield important information. We can regard graphs as one-dimensional simplicial complexes, and as whether there is a generalisation of the Laplacian operator to simplicial complexes. It turns out that there is, and that is useful for calculating real Betti numbers [8]. Duval and Reiner [5] have studied Laplacians of a sp...

We give new examples of shellable but not extendably shellable two dimensional simplicial complexes. They include minimal examples, which are smaller than those previously known. We also give examples of shellable but not vertex decomposable two dimensional simplicial complexes. Among them are extendably shellable ones. This shows that neither extendable shellability nor vertex decomposability ...

Simplicial complexes are a versatile and convenient paradigm on which to build all the tools techniques of logic knowledge, assumption that initial epistemic models can be described in distributed fashion. Thus, we define: belief, bisimulation, group notions mutual, common also dynamics shape simplicial action models. We give survey how interpret such complexes, building upon foundations laid G...

Some recent attempts of settling this question go by looking at the problem in the more general context of pure simplicial complexes: What is the maximum diameter of the dual graph of a simplicial (d − 1)sphere or (d− 1)-ball with n vertices? Here a simplicial (d − 1)-ball or sphere is a simplicial complex homeomorphic to the (d − 1)-ball or sphere. These complexes are necessarily pure (all the...

We introduce a large self-dual class of simplicial complexes about which we show that each complex in it is contractible or homotopy equivalent to a sphere. Examples of complexes in this class include independence and dominance complexes of forests, pointed simplicial complexes, and their combinatorial Alexander duals.

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from graph theory to simplicial complexes.