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One-dimensional abstract simplicial complexes are mathematically equivalent to simple undirected graphs: the vertex set of the complex can be viewed as the vertex set of a graph, and the two-element facets of the complex correspond to undirected edges of a graph. In this view, one-element facets of a complex correspond to isolated vertices that ...
Let be an abstract simplicial complex on a vertex set of size . The Alexander dual X ∗ {\displaystyle X^{*}} of X {\displaystyle X} is defined as the simplicial complex on V {\displaystyle V} whose faces are complements of non-faces of X {\displaystyle X} .
In algebraic combinatorics, the Kruskal–Katona theorem gives a complete characterization of the f-vectors of abstract simplicial complexes.It includes as a special case the Erdős–Ko–Rado theorem and can be restated in terms of uniform hypergraphs.
An abstract simplicial complex (ASC) is family of sets that is closed under taking subsets (the subset of a set in the family is also a set in the family). Every abstract simplicial complex has a unique geometric realization in a Euclidean space as a geometric simplicial complex (GSC), where each set with k elements in the ASC is mapped to a (k-1)-dimensional simplex in the GSC.
Let Δ be an abstract simplicial complex of dimension d − 1 with f i i-dimensional faces and f −1 = 1. These numbers are arranged into the f-vector of Δ, = (,, …,).An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.
The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset ( S , ≤) is then the Alexandrov topology on the order complex associated to ( S , ≤).
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A simplicial 3-complex. In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their n-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration).