Characterizations of Line Simplicial Complexes

TL;DR

This paper explores the properties of line simplicial complexes derived from graphs, establishing their connectedness, relationships with Euler characteristics, and shellability for different graph classes.

Contribution

It introduces and characterizes line and Gallai simplicial complexes, proving their connectedness criteria and analyzing their topological properties such as shellability.

Findings

Line simplicial complex is connected iff the graph is connected.

Euler characteristics relate line and Gallai simplicial complexes.

Shellability varies across different classes of graphs.

Abstract

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes $\Delta_{\Gamma}(G)$ and $\Delta_{\Gamma'}(G)$ (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects. In Theorem 3.3, we prove that the line simplicial complex $\Delta_L(G)$ is connected if and only if $G$ is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes. In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.

Equations

$$\chi (\Delta )=\sum _{k=0}^{n-1}(-1{)}^k{\alpha }_k.$$

$${\Delta }_L(G)=<F|F\in {\rm Y}(G)>,$$

$$\chi ({\Delta }_L(G))=\chi ({\Delta }_{\Gamma }(G))+\mid {\Omega }_{{\Gamma }^{\prime }}(G)\mid ,$$

$$\chi ({\Delta }_L(G))=\sum _{k=1}^t\chi ({\Delta }_L(G_k)).$$

$$\chi ({\Delta }_L(G_k))=\chi ({\Delta }_{\Gamma }(G_k))+\mid {\Omega }_{{\Gamma }^{\prime }}(G_k)\mid ,$$

$$\chi ({\Delta }_L(G))=\sum _{k=1}^t\chi ({\Delta }_{\Gamma }(G_k))+\sum _{k=1}^t\mid {\Omega }_{{\Gamma }^{\prime }}(G_k)\mid =\chi ({\Delta }_{\Gamma }(G))+\mid {\Omega }_{{\Gamma }^{\prime }}(G)\mid$$

$$\chi ({\Delta }_L(G))=\chi ({\Delta }_{\Gamma }(G))+\mid {\Omega }_{{\Gamma }^{\prime }}(G)\mid ,$$

$$\chi ({\Delta }_L(W_{n+1}))={\alpha }_0-{\alpha }_1+{\alpha }_2=(n+1)-\frac{n(n+1)}{2}+\frac{n(n+1)}{2}=n+1,$$

$${\Omega }_{{\Gamma }^{\prime }}(W_{n+1})=\{F_{1,2,n+1},F_{2,3,n+1},\dots ,F_{n,1,n+1}\}.$$

$$\chi ({\Delta }_{\Gamma }(W_{n+1}))=\chi ({\Delta }_L(W_{n+1}))-\mid {\Omega }_{{\Gamma }^{\prime }}(W_{n+1})\mid =n+1-n=1$$

$$E(F_n)=\{e_{1,2},e_{3,4},\dots ,e_{2n-1,2n},e_{1,2n+1},e_{2,2n+1},\dots ,e_{2n,2n+1}\},$$

$$\chi ({\Delta }_{\Gamma }(F_n))={\alpha }_0-{\alpha }_1+{\alpha }_2=(2n+1)-n(2n+1)+2n(n-1)=1-n,$$

$${\hat{\Delta }}_{<F_i>}=<F_1,F_2,\dots ,F_{i-1}>\cap <F_i>$$

$${{\rm Y}}_L(F_n)=\{F_{1,2,2n+1},\dots ,F_{1,2n,2n+1},F_{2,3,2n+1},\dots ,F_{2,2n,2n+1},\dots ,F_{2n-1,2n,2n+1}\},$$

$$\frac{m_{F_{1,j,2n+1}}}{gcd(m_{F_{1,j-1,2n+1}},m_{F_{1,j,2n+1}})}=x_j,3\le j\le 2n$$

$$\frac{m_{F_{i,j,2n+1}}}{gcd(m_{F_{1,i,2n+1}},m_{F_{i,j,2n+1}})}=x_j$$

$$\frac{m_{F_{i,j,2n+1}}}{gcd(m_{F_{1,j,2n+1}},m_{F_{i,j,2n+1}})}=x_i.$$

$$\frac{m_{F_{1,j,n+1}}}{gcd(m_{F_{1,j-1,n+1}},m_{F_{1,j,n+1}})}=x_j,3\le j\le n.$$

$$\frac{m_{F_{i,j,n+1}}}{gcd(m_{F_{1,i,n+1}},m_{F_{i,j,n+1}})}=x_j$$

$$\frac{m_{F_{i,j,n+1}}}{gcd(m_{F_{1,j,n+1}},m_{F_{i,j,n+1}})}=x_i.$$

$$\frac{m_{F_{i,j,k}}}{gcd(m_{F_{i,j,n+1}},m_{F_{i,j,k}})}=x_k,$$

$$\frac{m_{F_{i,j,k}}}{gcd(m_{F_{j,k,n+1}},m_{F_{i,j,k}})}=x_i$$

$$\frac{m_{F_{i,j,k}}}{gcd(m_{F_{i,k,n+1}},m_{F_{i,j,k}})}=x_j.$$

$${\Delta }_L(C_5)=⟨\{1,2,3\},\{2,3,4\},\{3,4,5\},\{4,5,1\},\{5,1,2\}⟩$$

$$I_{\mathcal{F}}({\Delta }_L(C_5))=(m_{F_{1,2,3}},m_{F_{2,3,4}},m_{F_{3,4,5}},m_{F_{4,5,1}},m_{F_{5,1,2}}).$$

$${\Omega }_{{\Gamma }^{\prime }}(W_{n+1})=\{F_{1,2,n+1},F_{2,3,n+1},\dots ,F_{n-1,n,n+1},F_{n,1,n+1}\},$$

$$I_{\mathcal{F}}({\Delta }_{{\Gamma }^{\prime }}(W_{n+1}))=(x_{F_{1,2,n+1}},x_{F_{2,3,n+1}},\dots ,x_{F_{n-1,n,n+1}},x_{F_{n,1,n+1}}),$$

$$\frac{m_{F_{i,i+1,n+1}}}{gcd(m_{F_{i-1,i,n+1}},m_{F_{i,i+1,n+1}})}=x_{i+1},2\le i\le n-1.$$

$${\Omega }_{{\Gamma }^{\prime }}(Y_{3,n})=\{F_{1,2,3},F_{4,5,6},\dots ,F_{3n-2,3n-1,3n}\}.$$

$$\frac{m_{F_{j,j+1,j+2}}}{gcd(m_{F_{i,i+1,i+2}},m_{F_{j,j+1,j+2}})}=x_j{x}_{j+1}{x}_{j+2},$$

Full text

Key words: Euler characteristic, simplicial complex, facet ideal, connected simplicial complex and Shellability.

*2010 Mathematics Subject Classification: Primary 05E25, 55U10, 13P10 Secondary 06A11, 13H10.

1. Introduction

Let $\Delta$ be a simplicial complex on the vertex set $[n]=\{1,\ldots,n\}$ and denote by $\alpha_{k}$ the number of $k$-cells of $\Delta$. Then, the Euler characteristic of the simplicial complex $\Delta$ is given by

[TABLE]

The Euler characteristic is a famous topological and homotopic invariant to classify surfaces, see [9] and [12]. The excision is one of the most useful property of Euler characteristic, given by $\chi(\Delta)=\chi(C)+\chi(\Delta\backslash C)$, for every closed subset $C\subset\Delta$. The excision property has a dual form $\chi(\Delta)=\chi(U)+\chi(\Delta\backslash U)$, for every open subset $U\subset\Delta$. This property is frequently used under the guise of the inclusion-exclusion formula.

The shellability of a simplical complex $\Delta$ is a well-studied combinatorial property that carries strong geometric and algebraic interpretations, see for example [13]. In many situations, proving shellability is the most efficient way of establishing Cohen-Macaulayness, see for instance [3]. The algebraic criterion for the shellability of a simplicial complex has been firstly introduced by A. Dress [5]. In [6], Eagon and Reiner gave algebraic criterion of the pure shellability of a dual simplicial complex $\check{\Delta}$ in the context of the Stanley-Reisner ideal theory.

Recently, in [2], Anwar, Kosar and Nazir gave a translation of the shellability of a simplicial complex $\Delta$ on the monomial generators of the facet ideal $I_{\mathcal{F}}(\Delta)$. Their algebraic translation provided an useful class of ideals known as ideals with Linear residuals.

Let $G$ be a finite simple graph. The line graph $L(G)$ of $G$ is a graph having edges of $G$ as its vertices and two distinct edges of $G$ are adjacent in $L(G)$ if they are adjacent in $G$. It was firstly introduced by Harary and Norman in [8].

Both the Gallai and anti-Gallai graphs $\Gamma(G)$ and $\Gamma^{\prime}(G)$ of a graph $G$ have the edges of $G$ as their vertices. Two edges of $G$ are adjacent in the Gallai graph $\Gamma(G)$ if they are incident but do not span a triangle in $G$; they are adjacent in the anti-Gallai graph $\Gamma^{\prime}(G)$ if they span a triangle in $G$, see [7] and [10]. The Gallai and anti-Gallai graphs are spanning subgraphs of the line graph $L(G)$. The anti-Gallai graph $\Gamma^{\prime}(G)$ is the complement of $\Gamma(G)$ in $L(G)$.

We define first the line simplicial complex $\Delta_{L}(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes $\Delta_{\Gamma}(G)$ and $\Delta_{\Gamma^{\prime}}(G)$ (respectively) as spanning subcomplexes. The study of connectedness of simplicial complexes is interesting due to various combinatorial and topological aspects, see [4] and [11]. In Theorem 3.3, we prove that the line simplicial complex $\Delta_{L}(G)$ is connected if and only if $G$ is connected. In Theorem 3.4, we establish the relation between Euler characteristics of line and Gallai simplicial complexes.

In Section 4, we discuss the shellability of line and anti-Gallai simplicial complexes associated to various classes of graphs.

2. Preliminaries

A simplicial complex $\Delta$ on the vertex set $[{n}]=\{1,\ldots,n\}$ is a subset of $2^{[n]}$ with the property that if $F\in\Delta$ then every subset of $F$ will belong to $\Delta$. The members of $\Delta$ are called faces and the maximal faces under inclusion are called facets. If $\mathcal{F}(\Delta)=\{F_{1},\ldots,F_{h}\}$ is the set of all facets of $\Delta$, then $\Delta=<F_{1},\ldots,F_{h}>$. A subcomplex of the simplicial complex $\Delta$ is a simplicial complex whose facet set is a subset of $\mathcal{F}(\Delta)$. The dimension of a face $F\in\Delta$ is given by $\dim F=\mid F\mid-1$, where $\mid F\mid$ is the number of vertices of $F$. The dimension of a simplical complex $\Delta$ is defined by $\dim\Delta=max\{\dim F\ |\ F\in\Delta\}$. A simplicial complex $\Delta$ is said to be pure if it has all facets of the same dimension.

A simplicial complex $\Delta$ is said to be connected if for any two facets $F$ and $\tilde{F}$ of $\Delta$, there exists a sequence of facets $F=F_{0},\ldots,F_{q}=\tilde{F}$ such that $F_{i}\cap F_{i+1}\neq\emptyset$ for any $i=0,\ldots,q-1$. A disconnected simplicial complex is a complex which is not connected. That is, the vertex set $[n]$ of $\Delta$ can be written as disjoint union of two non-empty subsets $V_{1}$ and $V_{2}$ of $[n]$ such that no face of $\Delta$ has vertices in both $V_{1}$ and $V_{2}$, see [4] and [11].

We define now the line graph $L(G)$, which provides the main streamline of this work, see [8].

Definition 2.1**.**

Let $G$ be a finite simple graph. The graph $L(G)$ is said to be line graph of $G$ if each vertex of $L(G)$ represents an edge of $G$ and two vertices of $L(G)$ are adjacent if and only if their corresponding edges are incident in $G$.

Example 2.2**.**

The graph $G$ and its Line graph $L(G)$ are given in Figure  1.

3. Topological Characterizations of Line Simplicial Complexes

The following definition plays a key role in the structural study of line graph $L(G)$.

Definition 3.1**.**

Let $G$ be a finite simple graph on the vertex set $V(G)=[n]$. Let $E(G)=\{e_{i,j}=\{i,j\}|i,j\in V(G)\}$ be the edge set of $G$. We define the set of line indices $\Upsilon(G)$ associated to the graph $G$ as the collection of subsets of $V(G)$ such that if $e_{i,j}$ and $e_{j,k}$ are adjacent in $L(G)$, then $F_{i,j,k}=\{i,j,k\}\in\Upsilon(G)$ or if $e_{i,j}$ is an isolated vertex in $L(G)$ then $F_{i,j}=\{i,j\}\in\Upsilon(G)$.

The line index is said to be a Gallai index if the incident edges $e_{i,j}$ and $e_{j,k}$ of $G$ do not span a triangle in $G$. We denote the set of Gallai indices by $\Omega_{\Gamma}(G)$, see [1] and [2]. The line index is said to be an anti-Gallai index if the incident edges $e_{i,j}$ and $e_{j,k}$ of $G$ lie on a triangle in $G$. We denote the set of anti-Gallai indices by $\Omega_{\Gamma^{\prime}}(G)$. The set of line indices $\Upsilon(G)$ contains $\Omega_{\Gamma}(G)$ and $\Omega_{\Gamma^{\prime}}(G)$ as spanning subsets.**

Definition 3.2**.**

A line simplicial complex $\Delta_{L}(G)$ of $G$ is a simplicial complex on the vertex set $V(G)$ such that

[TABLE]

where $\Upsilon(G)$ is the set of line indices of $G$.**

Similarly, the Gallai and anti-Gallai simplicial complexes $\Delta_{\Gamma}(G)$ and $\Delta_{\Gamma^{\prime}}(G)$ are generated by Gallai and anti-Gallai indices, respectively. We refer [1] and [2] for Gallai simplicial complex. The line simplicial complex $\Delta_{L}(G)$ contains Gallai and anti-Gallai simplicial complexes as spanning subcomplexes. The anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}}(G)$ is complement of $\Delta_{\Gamma}(G)$ in $\Delta_{L}(G)$.

We prove first necessary and sufficient condition for connectedness of the line simplicial complex $\Delta_{L}(G)$.

Theorem 3.3**.**

Let $G$ be a finite simple graph on the vertex set $[n]$. Then, $G$ is connected if and only if the line simplicial complex $\Delta_{L}(G)$ is connected.

Proof.

Let $G$ be a finite simple graph on the vertex set $[n]$. For $n\leq 3$, the result is trivial. Therefore, we take $n\geq 4$.

We establish first direct implication. On contrary, we assume that the line simplicial complex $\Delta_{L}(G)$ is not connected. By definition, there exists two non-empty subsets $V_{1}$ and $V_{2}$ of $[n]$ such that $[n]=V_{1}\cup V_{2}$ and $V_{1}\cap V_{2}=\emptyset$ with the property that any facet of $\Delta_{L}(G)$ either has vertices from $V_{1}$ or $V_{2}$. Since $G$ is connected graph on the vertex set $[n]$ with $n\geq 4$, therefore the line simplicial complex $\Delta_{L}(G)$ is pure of dimension $2$. So, there are facets, say $F_{i,j,k},F_{l,m,p}\in\Delta_{L}(G)$ for every $i\neq j\neq k\in V_{1}\subset[n]$ and for every $l\neq m\neq p\in V_{2}\subset[n]$ such that $e_{i,j},e_{j,k}$ and $e_{l,m},e_{m,p}$ are adjacent vertices of the line graph $L(G)$. Therefore, the edges $e_{i,j},e_{j,k}$ and $e_{l,m},e_{m,p}$ are incident in $G$ for every $i,j,k\in V_{1}$ and for every $l,m,p\in V_{2}$ with $[n]=V_{1}\cup V_{2}$ such that $V_{1}\cap V_{2}=\emptyset$, a contradiction.

Now, we prove converse implication. On contrary, we assume that the graph $G$ is not connected. That is, there exist two vertices, say $r,s\in G$ such that no path in $G$ has $r$ and $s$ as end points. It implies that there is no face of $\Delta_{L}(G)$ containing both vertices $r$ and $s$ i.e. $\Delta_{L}(G)$ is not connected, a contradiction. Hence the result.

We establish now the relation between Euler characteristics of line and Gallai simplicial complexes.

Theorem 3.4**.**

Let $\Delta_{L}(G)$ and $\Delta_{\Gamma}(G)$ be line and Gallai simplicial complexes of a finite simple graph $G$. Then, the Euler characteristic of the line simplicial complex $\Delta_{L}(G)$ is given by

[TABLE]

where $\mid\Omega_{\Gamma^{\prime}}(G)\mid$ is the number of anti-Gallai indices associated to $G$.

Proof.

Let $G$ be a finite simple graph consisting of $t$ connected components $G_{1},\ldots,G_{t}$. Then, $G=\cup_{k=1}^{t}\,\,G_{k}$ such that $G_{i}\cap G_{j}=\emptyset$ for every $i,j=1,\ldots,t$ with $i\neq j$. By Theorem 3.3, the line simplicial complex $\Delta_{L}(G)$ also consists of $t$ connected components $\Delta_{L}(G_{1}),\ldots,\Delta_{L}(G_{t})$. Therefore, the line simplicial complex can be expressed as $\Delta_{L}(G)=\cup_{k=1}^{t}\,\,\Delta_{L}(G_{k})$ such that $\Delta_{L}(G_{i})\cap\Delta_{L}(G_{j})=\emptyset$ for all $i,j=1,\ldots,t$ with $i\neq j$.

By the excision property, the Euler characteristic of the line simplicial complex $\Delta_{L}(G)$ is given by

[TABLE]

Let $\Delta_{\Gamma}(G_{k})$ and $\Delta_{\Gamma^{\prime}}(G_{k})$ be Gallai and anti-Gallai simplicial complexes associated to each connected component $G_{k}$ of $G$ for $k=1,\ldots,t$. Then, each connected component $\Delta_{L}(G_{k})$ of line simplicial complex contains $\Delta_{\Gamma}(G_{k})$ and $\Delta_{\Gamma^{\prime}}(G_{k})$ as spanning subcomplexes for $k=1,\ldots,t$. Therefore, by definition, the Euler characteristic of each connected component $\Delta_{L}(G_{k})$ of line simplicial complex is given by

[TABLE]

where $\Omega_{\Gamma^{\prime}}(G_{k})$ is the set of anti-Gallai indices associated to $G_{k}$ for $k=1,\ldots,t$. Consequently,

[TABLE]

due to excision property. Hence proved.

Example 3.5**.**

Consider the graph $G$ given in Figure  2.

Then $\Delta_{L}(G)=<\{1,2,3\},\{1,3,4\},\{2,3,4\}>$ and $\chi(\Delta_{L}(G))=f_{0}-f_{1}+f_{2}=4-6+3=1$, where $f_{i}$ is the $i$-dimensional faces of $\Delta_{L}(G)$.

Also, $\Delta_{\Gamma}(G)=<\{1,2\},\{1,3,4\},\{2,3,4\}>$ and $\chi(\Delta_{\Gamma}(G))=g_{0}-g_{1}+g_{2}=4-6+2=0$, where $g_{i}$ is the $i$-dimensional faces of $\Delta_{\Gamma}(G)$.

And $|\Omega_{\Gamma^{\prime}}(G)|=|\{1,2,3\}|=1$. Therefore,

[TABLE]

where $\mid\Omega_{\Gamma^{\prime}}(G)\mid$ is the number of anti-Gallai indices associated to $G$.

Example 3.6**.**

Let $W_{n+1}$ be the wheel graph on $n+1$ vertices having edge set $E(W_{n+1})=\{e_{1,2},\ldots,e_{n,1},e_{1,n+1},\ldots,e_{n,n+1}\},$ as shown in the Figure  3.

Then, the line indices of the wheel graph are given by

$\Upsilon(W_{n+1})=\{F_{1,2,3},\ldots,F_{n,1,2},F_{1,2,n+1},F_{1,3,n+1},\ldots,F_{1,n,n+1},F_{2,3,n+1},\ldots,\\ F_{2,n,n+1},\ldots,F_{n-1,n,n+1}\}$. We compute first the Euler characteristic of the line simplicial complex $\Delta_{L}(W_{n+1})$ for $n\geq 4$. There are $(n+1)$ 0-dimensional faces or vertices in $\Delta_{L}(W_{n+1})$ i.e. $\alpha_{0}=n+1$. Now, the number of $1$-dimensional faces of $\Delta_{L}(W_{n+1})$ is given by $\alpha_{1}=|\{j,k\}|={n+1\choose 2}=\frac{n(n+1)}{2}$, where $1\leq j\neq k\leq n+1$. Next, we compute the number of $2$-dimensional faces of the form $\{j,k,l\}\in\Delta_{L}(W_{n+1})$ with $1\leq j,k\leq n$ and $1\leq l\leq n+1$ such that $j\neq k\neq l$.

1. (1)

$|\{j,j+1,j+2\}|=n-2$, where $1\leq j\leq n-2$; 2. (2)

$|\{n-1,n,1\}|=1$; 3. (3)

$|\{n,1,2\}|=1$; 4. (4)

$|\{j,k,n+1\}|={n\choose 2}=\frac{n(n-1)}{2}$, where $1\leq j\neq k\leq n$.

Adding from $(1)$ to $(4)$, we get $\alpha_{2}=n-2+1+1+\frac{n(n-1)}{2}=\frac{n(n+1)}{2}$.

Thus, we obtain

[TABLE]

where $n\geq 4$. By definition, the anti-Gallai indices of the wheel graph $W_{n+1}$ are given by

[TABLE]

It implies that $\mid\Omega_{\Gamma^{\prime}}(W_{n+1})\mid=n$. Therefore, by Theorem 3.4,

[TABLE]

with $n\geq 4$.

Example 3.7**.**

Let $F_{n}$ be Friendship graph on $2n+1$ vertices with edge set

[TABLE]

as shown in the Figure  4.

Then, the Gallai indices of $F_{n}$ are given by

$\Omega_{\Gamma}(F_{n})=\{F_{1,2},\ldots,F_{2n-1,2n},F_{1,3,2n+1},\ldots,F_{1,2n,2n+1},F_{2,3,2n+1},\ldots,F_{2,2n,2n+1},\\ F_{3,5,2n+1},\ldots,F_{2n-2,2n,2n+1}\}$, see [2]. We compute first the Euler characteristic of the Gallai simplicial complex $\Delta_{\Gamma}(F_{n})$ for $n\geq 2$. Since, the friendship graph $F_{n}$ has $(2n+1)$ vertices, therefore the number of [math]-dimensional faces in $\Delta_{\Gamma}(F_{n})$ is $\alpha_{0}=2n+1$. Moreover, the number of $1$-dimensional faces of $\Delta_{\Gamma}(F_{n})$ is given by $\alpha_{1}=|\{i,j\}|={2n+1\choose 2}=n(2n+1)$, where $1\leq i\neq j\leq 2n+1$. Now, we compute the number of $2$-dimensional faces of the form $\{i,j,k\}\in\Delta_{\Gamma}(F_{n})$ with $1\leq i\leq 2n-2$, $3\leq j\leq 2n$ and $k=2n+1$ such that $i\neq j$.

(1)      $|\{i,j,2n+1\}|=2(2n-2)$ with $i\in\{1,2\}$ and $3\leq j\leq 2n$;

(2)      $|\{i,j,2n+1\}|=2(2n-4)$ with $i\in\{3,4\}$ and $5\leq j\leq 2n$;

$\vdots$

(n-2)  $|\{i,j,2n+1\}|=2(4)$ with $i\in\{2n-5,2n-4\}$ and $2n-3\leq j\leq 2n$;

(n-1)  $|\{i,j,2n+1\}|=2(2)$ with $i\in\{2n-3,2n-2\}$ and $2n-1\leq j\leq 2n$.

Adding from $(1)$ to $(n-1)$, we obtain

$\alpha_{2}=2(2n-2)+2(2n-4)+\cdots+2(4)+2(2)=2n(n-1)$.

Therefore, we compute

[TABLE]

where $n\geq 2$. Note that $\mid\Omega_{\Gamma^{\prime}}(F_{n})\mid=n$. Hence, by Theorem 3.4, $\chi(\Delta_{L}(F_{n}))=1$ with $n\geq 2$.

Remark 3.8**.**

Let $G$ be a finite simple graph. If there is no triangle in $G$, then there will be no anti-Gallai index in $\Upsilon(G)$ i.e. $\Omega_{\Gamma^{\prime}}(G)=\emptyset$ and the line and Gallai simplicial complexes of $G$ are coincident.

It can be easily seen that the line simplicial complexes associated to friendship graph $F_{n}$ and star graph $S_{2n}$ are the same.

4. Shellability of Line and Anti-Gallai Simplicial Complexes

We introduce first a few notions.

Definition 4.1**.**

A simplicial complex $\Delta$ over $[n]$ is shellable if its facets can be ordered $F_{1},F_{2},\ldots,F_{s}$ such that, for all $2\leq i\leq s$ the subcomplex

[TABLE]

is pure of dimension $dim(F_{i})-1$.

Definition 4.2**.**

Let $I\subset S=k[x_{1},\ldots,x_{n}]$ be a monomial ideal. We say that $I$ has Linear Residuals if there exist an ordered minimal monomial system of generators $\{m_{1},m_{2},\ldots,m_{r}\}$ of $I$ such that Res$(I_{i})$ is minimally generated by linear monomials for $1<i\leq r$, where Res$(I_{i})=\{u_{1},u_{2},\ldots,u_{i-1}\}$ such that $u_{k}=\frac{m_{i}}{gcd(m_{k},m_{i})}$ for all $1\leq k\leq i-1$.

The following result provides effective necessary and sufficient condition for the shellability of a simplicial complex, see [2].

Theorem 4.3**.**

[2*]** *Let $\Delta$ be a simplicial complex of dimension $d$ over $[n]$. Then $\Delta$ will be shellable if and only if $I_{\mathcal{F}}(\Delta)$ has linear residuals.

Theorem 4.4**.**

The line simplicial complex $\Delta_{L}(F_{n})$ associated to friendship graph $F_{n}$ is shellable.

Proof.

By Theorem 4.3, it is sufficient to show that $I_{\mathcal{F}}(\Delta_{L}(F_{n}))$ have linear residuals. As the line indices of the friendship graph $F_{n}$ on $2n+1$ vertices are given by

[TABLE]

as shown in Figure  4. Then, the ordered minimal monomial system of generators are given by

$I_{\mathcal{F}}(\Delta_{L}(F_{n}))=(m_{F_{1,2,2n+1}},\ldots,m_{F_{1,2n,2n+1}},m_{F_{2,3,2n+1}},\ldots,m_{F_{2,2n,2n+1}},\\ \ldots,m_{F_{2n-1,2n,2n+1}})$, where $m_{F_{i,j,2n+1}}$ are the monomial $x_{i}x_{j}x_{2n+1}$. One can easily see that Res$(I_{m_{F_{1,j,2n+1}}})$ is minimally generated by

[TABLE]

Moreover, Res$(I_{m_{F_{i,j,2n+1}}})$ with $1<i<j\leq 2n$ is minimally generated by the linear monomials $x_{j}$ and $x_{i}$ due to

[TABLE]

and

[TABLE]

Theorem 4.5**.**

The line simplicial complex $\Delta_{L}(W_{n+1})$ associated to wheel graph $W_{n+1}$ is shellable.

Proof.

The ordered minimal monomial system of generators are given by

$I_{\mathcal{F}}(\Delta_{L}(W_{n+1}))=(m_{F_{1,2,n+1}},\ldots,m_{F_{1,n,n+1}},m_{F_{2,3,n+1}},\ldots,m_{F_{2,n,n+1}},\ldots,m_{F_{n-1,n,n+1}},\\ m_{F_{1,2,3}},\ldots,m_{F_{n,1,2}})$, where $m_{F_{i,j,k}}$ are the monomials $x_{i}x_{j}x_{k}$ associated to the facets $F_{i,j,k}$, see Figure  3. We establish the result into the following steps.

Step.I. For the monomials $m_{F_{1,2,n+1}},\ldots,m_{F_{1,n,n+1}}$, one can easily see that Res$(I_{m_{F_{1,j,n+1}}})$ is minimally generated by

[TABLE]

Step.II. For the monomials $m_{F_{2,3,n+1}},\ldots,m_{F_{2,n,n+1}},\ldots,m_{F_{n-1,n,n+1}}$, Res$(I_{m_{F_{i,j,n+1}}})$ with $1<i<j\leq n$ is minimally generated by the linear monomials $x_{i}$ and $x_{j}$ due to

[TABLE]

and

[TABLE]

Step.III. For the monomials $m_{F_{1,2,3}},\ldots,m_{F_{n-2,n-1,n}}$, Res$(I_{m_{F_{i,j,k}}})$ with $1\leq i<j<k\leq n$ is minimally generated by the linear monomials $x_{i},x_{j}$ and $x_{k}$ due to

[TABLE]

[TABLE]

and

[TABLE]

Step.IV. Finally, one can easily see that the residuals

Res$(I_{m_{F_{n-1,n,1}}})=\{x_{1},x_{n-1},x_{n}\}$ and Res$(I_{m_{F_{n,1,2}}})=\{x_{1},x_{2},x_{n}\}$

are minimally generated by linear monomials.

Example 4.6**.**

Consider the line simplicial complex

[TABLE]

associated to the cycle $C_{5}=(12345)$ on $5$ vertices. Then

[TABLE]

On contrary, we assume that $\Delta_{L}(C_{5})$ is shellable. Therefore, by Theorem 4.3, $I_{\mathcal{F}}(\Delta_{L}(C_{5}))$ admits linear residuals. Without loss of generality, we may assume that $\mathbf{m_{1}}=m_{F_{1,2,3}}$ and $\mathbf{m_{2}}=m_{F_{2,3,4}}$. If we take $\mathbf{m_{3}}=m_{F_{4,5,1}}$, then we have Res$(I_{\mathbf{m_{3}}})=\{x_{4}x_{5},x_{1}x_{5}\}$ not generated by linear monomials. Therefore, $\mathbf{m_{3}}\neq m_{F_{4,5,1}}$. So, we have either $\mathbf{m_{3}}=m_{F_{5,1,2}}$ or $\mathbf{m_{3}}=m_{F_{3,4,5}}$. If $\mathbf{m_{3}}=m_{F_{5,1,2}}$, we have either $\mathbf{m_{4}}=m_{F_{4,5,1}}$ or $\mathbf{m_{4}}=m_{F_{3,4,5}}$. Then, the residuals

$Res(I_{\mathbf{m_{4}}})=Res(I_{m_{F_{4,5,1}}})=\{x_{1}x_{5},x_{4}\}$ and $Res(I_{\mathbf{m_{4}}})=Res(I_{m_{F_{3,4,5}}})=\{x_{5},x_{3}x_{4}\}$

are not minimally generated by linear monomials.

If $\mathbf{m_{3}}=m_{F_{3,4,5}}$, then either $\mathbf{m_{4}}=m_{F_{4,5,1}}$ or $\mathbf{m_{4}}=m_{F_{5,1,2}}$. Then, the residuals

$Res(I_{\mathbf{m_{4}}})=Res(I_{m_{F_{4,5,1}}})=\{x_{1},x_{4}x_{5}\}$ and $Res(I_{\mathbf{m_{4}}})=Res(I_{m_{F_{5,1,2}}})=\{x_{5},x_{1}x_{2}\}$

are not minimally generated by linear monomials, which is a contradiction.

Theorem 4.7**.**

The anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}}(W_{n+1})$ associated to wheel graph $W_{n+1}$ is shellable.

Proof.

The anti-Gallai indices of wheel graph $W_{n+1}$ are given by

[TABLE]

as shown in Figure  5.

Then, the facet ideal $I_{\mathcal{F}}(\Delta_{\Gamma^{\prime}}(W_{n+1}))$ is given by

[TABLE]

where $x_{F_{i,j,n+1}}$ are the monomials $x_{i}x_{j}x_{n+1}$. It can be easily seen that Res$(I_{x_{F_{i,i+1,n+1}}})$ is minimally generated by

[TABLE]

Moreover, Res$(I_{x_{F_{n,1,n+1}}})=\{x_{1},x_{n}\}$. Thus, the anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}}(W_{n+1})$ is shellable.

In the following examples, we see that the anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}(G)}$ are not shellable.

Example 4.8**.**

Let $G=Y_{3,n}$ be prism graph having $3n$ vertices and $3(2n-1)$ edges, as shown in [1]. The anti-Gallai indices associated to the prism graph $Y_{3,n}$ are given by

[TABLE]

The anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}}(Y_{3,n})$ consisting of $n$ disjoint facets is pure of dimension $2$, as shown in Figure  6.

Note that Res$(I_{x_{F_{j,j+1,j+2}}})$ is minimally generated by monomials

[TABLE]

where $i,j\in\{1,4,\ldots,3n-2\}$ and $i<j$.Therefore, the facet ideal $I_{\mathcal{F}}(\Delta_{\Gamma^{\prime}}(Y_{3,n}))$ does not have linear residuals for any monomial ordering of minimal system of generators of $I_{\mathcal{F}}(\Delta_{\Gamma^{\prime}}(Y_{3,n}))$. Hence $\Delta_{\Gamma^{\prime}}(Y_{3,n})$ is not shellable.

Example 4.9**.**

The anti-Gallai indices associated to friendship graph $G=F_{n}$ are given by

[TABLE]

So, the anti-Gallai simplicial complex $\Delta_{\Gamma^{\prime}}(F_{n})$ consisting of $n$ facets with a common vertex is pure of dimension $2$, as shown in Figure  7.

The residual Res$(I_{x_{F_{j,j+1,2n+1}}})$ is minimally generated by monomials

[TABLE]

where $i,j\in\{1,3,\ldots,2n-1\}$ and $i<j$. Therefore, the facet ideal $I_{\mathcal{F}}(\Delta_{\Gamma^{\prime}}(F_{n})$ does not have linear residuals for any monomial ordering of minimal system of generators of $I_{\mathcal{F}}(\Delta_{\Gamma^{\prime}}(F_{n})$. Hence $\Delta_{\Gamma^{\prime}}(F_{n})$ is not shellable.