Spanning Simplicial Complex of Wheel Graph Wn

2019 ◽  
Vol 26 (02) ◽  
pp. 309-320 ◽  
Author(s):  
A. Zahid ◽  
M.U. Saleem ◽  
A. Kashif ◽  
M. Khan ◽  
M.A. Meraj ◽  
...  
Keyword(s):  
Simplicial Complex ◽  
Hilbert Series ◽  
Associated Primes ◽  
Face Ring ◽  
Combinatorial Properties ◽  
Ideal Formula ◽  
Wheel Graph ◽  
Vertex Set ◽  
Facet Ideal

In this paper, we explore the spanning simplicial complex of wheel graph Wn on vertex set [n]. Combinatorial properties of the spanning simplicial complex of wheel graph are discussed, which are then used to compute the f-vector and Hilbert series of face ring k[Δs(Wn)] for the spanning simplicial complex Δs(Wn). Moreover, the associated primes of the facet ideal [Formula: see text] are also computed.

scholarly journals On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m

2018 ◽  
Vol 16 (1) ◽  
pp. 250-259
Author(s):  
Zahid Raza ◽  
Agha Kashif ◽  
Imran Anwar
Keyword(s):  
Simplicial Complex ◽  
Hilbert Series ◽  
Current Work ◽  
Algebraic Characterization ◽  
Face Ring ◽  
The Face

AbstractIn this paper, some algebraic and combinatorial characterizations of the spanning simplicial complex Δs(𝓙n,m) of the Jahangir’s graph 𝓙n,m are explored. We show that Δs(𝓙n,m) is pure, present the formula for f-vectors associated to it and hence deduce a recipe for computing the Hilbert series of the Face ring k[Δs(𝓙n,m)]. Finally, we show that the face ring of Δs(𝓙n,m) is Cohen-Macaulay and give some open scopes of the current work.

scholarly journals A Construction of Sequentially Cohen–Macaulay Graphs

2021 ◽  
Vol 28 (03) ◽  
pp. 399-414
Author(s):  
Aming Liu ◽  
Tongsuo Wu
Keyword(s):  
Simplicial Complex ◽  
Betti Numbers ◽  
Simple Graph ◽  
Ideal Formula ◽  
Vertex Decomposable ◽  
Linear Quotients ◽  
Facet Ideal

For every simple graph [Formula: see text], a class of multiple clique cluster-whiskered graphs [Formula: see text] is introduced, and it is shown that all such graphs are vertex decomposable; thus, the independence simplicial complex [Formula: see text] is sequentially Cohen–Macaulay. The properties of the graphs [Formula: see text] and [Formula: see text] constructed by Cook and Nagel are studied, including the enumeration of facets of the complex [Formula: see text] and the calculation of Betti numbers of the cover ideal [Formula: see text]. We also prove that the complex[Formula: see text] is strongly shellable and pure for either a Boolean graph [Formula: see text] or the full clique-whiskered graph [Formula: see text] of [Formula: see text], which is obtained by adding a whisker to each vertex of [Formula: see text]. This implies that both the facet ideal [Formula: see text] and the cover ideal [Formula: see text] have linear quotients.

scholarly journals The facet ideal of a simplicial complex

2002 ◽  
Vol 109 (2) ◽  
pp. 159-174 ◽  
Author(s):  
Sara Faridi
Keyword(s):  
Simplicial Complex ◽  
Facet Ideal

scholarly journals Hard Squares with Negative Activity and Rhombus Tilings of the Plane

10.37236/1093 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Partition Function ◽  
Statistical Mechanics ◽  
Simplicial Complex ◽  
Transfer Matrix ◽  
Euler Characteristic ◽  
Independent Sets ◽  
Roots Of Unity ◽  
Hard Squares ◽  
Vertex Set

Let $S_{m,n}$ be the graph on the vertex set ${\Bbb Z}_m \times {\Bbb Z}_n$ in which there is an edge between $(a,b)$ and $(c,d)$ if and only if either $(a,b) = (c,d\pm 1)$ or $(a,b) = (c \pm 1,d)$ modulo $(m,n)$. We present a formula for the Euler characteristic of the simplicial complex $\Sigma_{m,n}$ of independent sets in $S_{m,n}$. In particular, we show that the unreduced Euler characteristic of $\Sigma_{m,n}$ vanishes whenever $m$ and $n$ are coprime, thereby settling a conjecture in statistical mechanics due to Fendley, Schoutens and van Eerten. For general $m$ and $n$, we relate the Euler characteristic of $\Sigma_{m,n}$ to certain periodic rhombus tilings of the plane. Using this correspondence, we settle another conjecture due to Fendley et al., which states that all roots of $\det (xI-T_m)$ are roots of unity, where $T_m$ is a certain transfer matrix associated to $\{\Sigma_{m,n} : n \ge 1\}$. In the language of statistical mechanics, the reduced Euler characteristic of $\Sigma_{m,n}$ coincides with minus the partition function of the corresponding hard square model with activity $-1$.

scholarly journals On the Facet Ideal of an Expanded Simplicial Complex

2018 ◽  
Vol 44 (3) ◽  
pp. 719-727
Author(s):  
S. Moradi ◽  
R. Rahmati-Asghar
Keyword(s):  
Simplicial Complex ◽  
Facet Ideal

scholarly journals The homotopy theory of polyhedral products associated with flag complexes

2018 ◽  
Vol 155 (1) ◽  
pp. 206-228
Author(s):  
Taras Panov ◽  
Stephen Theriault
Keyword(s):  
Simplicial Complex ◽  
Homotopy Theory ◽  
Chordal Graph ◽  
Polyhedral Product ◽  
Flag Complex ◽  
Vertex Set ◽  
Flag Complexes

If $K$ is a simplicial complex on $m$ vertices, the flagification of $K$ is the minimal flag complex $K^{f}$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\stackrel{}{\longrightarrow }K\stackrel{}{\longrightarrow }K^{f}$. This induces a sequence of maps of polyhedral products $(\text{}\underline{X},\text{}\underline{A})^{L}\stackrel{g}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K}\stackrel{f}{\longrightarrow }(\text{}\underline{X},\text{}\underline{A})^{K^{f}}$. We show that $\unicode[STIX]{x1D6FA}f$ and $\unicode[STIX]{x1D6FA}f\circ \unicode[STIX]{x1D6FA}g$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\text{}\underline{CY},\text{}\underline{Y})^{K}$ is a co-$H$-space if and only if the 1-skeleton of $K$ is a chordal graph, and we deduce that the maps $f$ and $f\circ g$ have right homotopy inverses in this case.

Cohen–Macaulay modifications of the facet ideal of a simplicial complex

2019 ◽  
Vol 19 (03) ◽  
pp. 2050060
Author(s):  
Safyan Ahmad ◽  
Imran Anwar ◽  
Zunaira Kosar
Keyword(s):  
Simplicial Complex ◽  
Definition Of ◽  
Facet Ideal

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.

On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

2016 ◽  
Vol 08 (03) ◽  
pp. 1650043 ◽  
Author(s):  
S. Visweswaran ◽  
Patat Sarman
Keyword(s):  
Commutative Ring ◽  
Discrete Math ◽  
Commutative Rings ◽  
Simple Graph ◽  
Integral Domains ◽  
Ideal Formula ◽  
Graph Theoretic ◽  
Vertex Set

The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced.

scholarly journals The Noncrossing Bond Poset of a Graph

10.37236/9253 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
C. Matthew Farmer ◽  
Joshua Hallam ◽  
Clifford Smyth
Keyword(s):  
Sufficient Conditions ◽  
Partition Lattice ◽  
Open Problems ◽  
Combinatorial Properties ◽  
Noncrossing Partition ◽  
Vertex Set ◽  
Connected Subgraphs ◽  
Necessary And Sufficient ◽  
Broken Circuit ◽  
Noncrossing Partition Lattice

The partition lattice and noncrossing partition lattice are well studied objects in combinatorics. Given a graph $G$ on vertex set $\{1,2,\dots, n\}$, its bond lattice, $L_G$, is the subposet of the partition lattice formed by restricting to the partitions whose blocks induce connected subgraphs of $G$. In this article, we introduce a natural noncrossing analogue of the bond lattice, the noncrossing bond poset, $NC_G$, obtained by restricting to the noncrossing partitions of $L_G$. Both the noncrossing partition lattice and the bond lattice have many nice combinatorial properties. We show that, for several families of graphs, the noncrossing bond poset also exhibits these properties. We present simple necessary and sufficient conditions on the graph to ensure the noncrossing bond poset is a lattice.  Additionally, for several families of graphs, we give combinatorial descriptions of the Möbius function and characteristic polynomial of the noncrossing bond poset. These descriptions are in terms of a noncrossing analogue of non-broken circuit (NBC) sets of the graphs and can be thought of as a noncrossing version of Whitney's NBC theorem for the chromatic polynomial. We also consider the shellability and supersolvability of the noncrossing bond poset, providing sufficient conditions for both. We end with some open problems. 

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