Vertex Decomposability of Path Complexes Associated to Cycle Graphs
Subject Areas : Multimedia Processing, Communications Systems, Intelligent SystemsSeyyed Mohammad Ajdani 1 , Kamal Ahmadi 2 , Asghar Madadi 3
1 - Assistant Professor, Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran
2 - Assistant Professor, Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran
3 - Assistant Professor, Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran
Keywords: Shellable, Matroid, Vertex decomposable, Cohen-Macaulay,
Abstract :
Introduction: Monomials are the link between commutative algebra and combinatorics. With a simplicial complex ∆, one can associate two square-free monomial ideals: the Stanley-Reisner ideal I_∆, whose generators correspond to the non-faces of ∆, or the facet ideal I(∆), which is a generalization of edge ideals of graphs and whose generators correspond to the facets of ∆. The facet ideal of a simplicial complex was first introduced by Faridi. Let G be a simple graph. The edge ideal I(G) of a graph G was first considered by R. Villarreal. He studied the algebraic properties of I(G) using a combinatorial language of G.
Method: In combinatorial commutative algebra, one can attach a monomial ideal to a combinatorial object. Then this ideal's algebraic properties are studied using the combinatorial properties of the combinatorial object. One of the interesting problems in combinatorial commutative algebra is the vertex decomposability of simplicial complexes, which many researchers study. In this abstract, we recall some definitions which will be needed later. A simplicial complex ∆ over a set of vertices V={x_1, … ,x_n} is a collection of subsets of V , with the property that {x_i }∊∆, for all i and if F∊∆, then all subsets of F are also in ∆ (including the empty set). An element in ∆ is called a face of ∆. The dimension of a face F of ∆, dimF, is |F|-1 where |F| is the number of elements of F. The maximal faces of ∆ under inclusion are called facets of ∆. The dimension of the simplicial complex ∆, dim∆, is the maximum of dimensions of its facets. If all facets of ∆ have the same dimension, then ∆ is called pure. Let Ƒ(∆)={F_1, … , F_q } be the facet set of ∆. It is clear that Ƒ(∆) determines ∆ completely and we write ∆=.
Results: Let G be a simple graph, and ∆_t (G) be a simplicial complex whose facets correspond to the paths of length t in G (t≥2). We show that ∆_t (C_n ) is matroid, vertex decomposable, shellable, and Cohen-Macaulay if and only if n=t or n=t+1, where C_n is an n-cycle.
