Abstract simplicial complexes and group presentations

Acta Mathematica Hungarica, 1992

We easily conclude that the incidence structure of 2 and its flag structure determine each other. Fig. 1 shows two 1-adjacent flags l//k(vj, er, flk) and 2//k(Vj, es, f/k), eT es, symbolically and with picture as well (d = 3). Note that the polyhedron 2 can be given to a computer as the set of its flags, i.e. ordered d-tuples of natural numbers. For instance a 3dimensional tetrahedron is represented by 24 flags, a cube by 48 flags. The one-to-one correspondence between an n-face x ~ and the flag subset :~x-, defined above, makes possible the convenient translation of geometric facts into flag language, i.e. computer one, and vice versa. The main purpose of this section is to describe an algorithm for determining the automorphism group of 2, denoted by Aut 2. This automatically describes the automorphism group Aut ~ of the flag structure 2~. Ant 2 is a permutation group. Each permutation in Aut 2 consists of d component permutations. The n-th component permutes the n-faces of 2, 0 _< n _< d-1. Every elements of Aut 2 preserves the incidence structure of 2, i.e. if an n-face x ~ and an m-face y'~ are incident then for each permutation p from Aut2 the n-th and m-th components of p maps x n onto (x'~)P~ ym onto (ym)Pso that the images are also incident. Aut~ consists of all self-bijections of ~ which preserve n-adjacency relations for every n (0 _< n _< d-1). These bijections form a group by the composition as group operation. LEMMA 1.1. An automorphism a from AutO, resp. Aut2, is uniquely determined (if it exists) by a flag ~ and its a-image [a.

Journal of Algebraic Combinatorics, 2006

Discrete Mathematics, 1994

Let E" be n-dimensional Euclidean space. A molecular space is a family of unit cubes in E". Any molecular space can be represented by its intersection graph. Conversely, it is known that any graph G can be represented by molecular space M(G) in E" for some n. Suppose that S, and S, are topologically equivalent surfaces in E" and molecular spaces M, and M, are the two families of unit cubes intersecting S, and S,, respectively. It was revealed that M, and M, could be transferred from one to the other with four kinds of contractible transformations if a division was small enough. In this paper, we will introduce the generating polynomial E,(x) and the Euler characteristic e(G) of a graph G. We will study several various operations performing on two graphs (surfaces). The generating polynomial of the new graph, which is obtained by performing various operations on well-studied graphs, can be expressed in terms of those of the old graphs. An immediate consequence is that the four contractible transformations do not change the Euler characteristic of a graph. Furthermore, we prove that all chordal graphs are contractible.

Discrete Applied Mathematics, 2004

We investigate the class of graphs deÿned by the property that every induced subgraph has a vertex which is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set). In particular we give the list of minimal forbidden subgraphs for the subclass of graphs whose vertex-set can be emptied out by ÿrst recursively eliminating simplicial vertices and then recursively eliminating co-simplicial vertices.

arXiv (Cornell University), 2021

Given an arbitrary hypergraph H, we may glue to H a family of hypergraphs to get a new hypergraph H ′ having H as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph H ′ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems. introduction A simplicial complex ∆ on a vertex set V is a collection of subsets of V such that ∪∆ = V and ∆ is closed under the operation of taking subsets. The elements of ∆ are called faces and the maximal faces of ∆, under inclusion, are called the facets of ∆. A simplicial complex with facets F 1 ,. .. , F m is often denoted by F 1 ,. .. , F m. A simplex is a simplicial complex with only one facet. A simplicial complex ∆ is called shellable if there is a total order on facets of ∆, say F 1 ,. .. , F m , such that F 1 ,. .. , F i−1 ∩ F i is generated by a non-empty set of maximal proper subsets of F i for 2 ≤ i ≤ m. The notion of shellability is used to give (an inductive) proof for the Euler-Poincaré formula in any dimension. If f i denotes the number of ifaces of a d-dimensional polytope (with f −1 = f d = 1), then the Euler-Poincaré formula states that d i=−1 (−1) i f i = 1. Shellable complexes are themselves an intermediate family among two other important families of simplicial complexes, namely vertex-decomposable and sequentially Cohen-Macaulay simplicial complexes. Indeed, we have the following implications vertex-decomposable =⇒ shellable =⇒ sequentially Cohen-Macaulay, and both of these inclusions are known to be strict. A vertex-decomposable simplicial complex ∆ is defined recursively in terms of link and deletion of vertices of ∆. In a more general setting, the link and the deletion of a face F of ∆ are defined as follows: