Certain Algebraic Invariants of Edge Ideals of Join of Graphs

TL;DR

This paper investigates algebraic invariants of edge ideals of graph joins, proving conjectures for specific graph classes, and exploring relationships between algebraic properties like regularity, depth, and multiplicity.

Contribution

It introduces new results on the regularity of symbolic powers of edge ideals for join graphs and proves Minh's conjecture for several graph classes.

Findings

Proves Minh's conjecture for wheel, complete multipartite, and certain co-chordal graphs.

Identifies classes of graphs with edge ideals having regularity three.

Shows multiplicity of edge ideals is independent of depth, dimension, regularity, and h-polynomial degree.

Abstract

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of $h$-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of $h$-polynomial by constructing graphs.

Equations

$$I(G{)}^{(s)}=\bigcap _{\mathfrak{p}\in \text{Ass}(I(G))}{\mathfrak{p}}^s.$$

$$\mathrm{reg}(I(G{)}^{(s)})=\mathrm{reg}(I(G{)}^s)\text{ for all }s\ge 1.$$

$$\text{dim}(\Delta )=\max \{|F|-1:F\text{ is a facet in }\Delta \}.$$

$$\mathrm{reg}\left((I+⟨y_1,\dots ,y_n⟩{)}^{(s)}+(J+⟨x_1,\dots ,x_m⟩{)}^{(s)}\right)=2s-1,\text{ for all }s\ge 1.$$

$$\mathrm{reg}(K)=\max \{j+1:(S/K{)}_j\ne 0\}.$$

$$\mathrm{reg}(I(G{)}^{(s)})=\max \{\mathrm{reg}\left(I(G_j{)}^{(i)}\right)-i+s:1\le i\le s,1\le j\le r\},\text{ for all }s\ge 1.$$

$$0\to I(G{)}^{(s)}\to (I(G_1)+{\mathfrak{m}}_2{)}^{(s)}\oplus (I(G_2)+{\mathfrak{m}}_1{)}^{(s)}\to (I(G_1)+{\mathfrak{m}}_2{)}^{(s)}+(I(G_2)+{\mathfrak{m}}_1{)}^{(s)}\to 0.$$

$$\mathrm{reg}((I(G_2)+{\mathfrak{m}}_1{)}^{(s)})=\max \{\mathrm{reg}(I(G_2{)}^{(i)})-i+s:1\le i\le s\}\ge 2s.$$

$$\mathrm{reg}((I(G_1)+{\mathfrak{m}}_2{)}^{(s)})=\max \{\mathrm{reg}(I(G_1{)}^{(i)})-i+s:1\le i\le s\}\ge 2s.$$

$$\mathrm{reg}\left((I(G_1)+{\mathfrak{m}}_2{)}^{(s)}+(I(G_2)+{\mathfrak{m}}_1{)}^{(s)}\right)<\max \{\mathrm{reg}((I(G_1)+{\mathfrak{m}}_2{)}^{(s)}),\mathrm{reg}((I(G_2)+{\mathfrak{m}}_1{)}^{(s)})\},$$

$$\mathrm{reg}\left(I(G{)}^{(s)}\right)=\max \left\{\mathrm{reg}\left((I(G_1)+{\mathfrak{m}}_2{)}^{(s)}\right),\mathrm{reg}\left((I(G_2)+{\mathfrak{m}}_1{)}^{(s)}\right)\right\}.$$

$$\mathcal{A}=\{G\mid \mathrm{reg}\left(I(G{)}^{s+1}:u\right)\le \mathrm{reg}\left(I(G)\right),u\in \mathcal{G}(I(G{)}^s),s\ge 1\},$$

$${\mathcal{A}}_1=\{G\mid \mathrm{reg}\left(I(G{)}^s\right)=2s+\mathrm{reg}(I(G))-2,s\ge 1\}.$$

$$\mathrm{reg}\left(I(G_i{)}^{(s)}\right)=\mathrm{reg}(I(G_i{)}^s)<\mathrm{reg}\left(I(G_i{)}^{s+1}\right)=\mathrm{reg}\left(I(G_i{)}^{(s+1)}\right).$$

$$\mathrm{reg}\left(I(G{)}^{(s)}\right)=\max \left\{\mathrm{reg}\left(I(G_i)\right):1\le i\le r\right\}+2s-2=2s+\mathrm{reg}(I(G))-2.$$

$$\mathrm{reg}(I(G{)}^{(s)})=\mathrm{reg}(I(G{)}^s)=2s+\nu (C_n)-1\text{ for all }s\ge 2.$$

$$0\to \frac{S}{I(G\ast H)}\to \frac{S}{I(G)+{\mathfrak{m}}_2}\oplus \frac{S}{I(H)+{\mathfrak{m}}_1}\to \mathsf{k}\to 0,$$

$$e(S/I(G\ast H))=\left\{\begin{array}[]{cc}e(S/I(G))+e(S/I(H))&\text{ if }m+\text{\rm ht}(I(H))=n+\text{\rm ht}(I(G)),\\ e(S/I(G))&\text{ if }m+\text{\rm ht}(I(H))>n+\text{\rm ht}(I(G)),\\ e(S/I(H))&\text{ if }m+\text{\rm ht}(I(H))<n+\text{\rm ht}(I(G)).\end{array}\right.$$

$$|\text{\rm Minh}(I(G\ast H))|=\left\{\begin{array}[]{cc}|\text{\rm Minh}(I(G))|+|\text{\rm Minh}(I(H))|&\text{ if }m+\text{\rm ht}(I(H))=n+\text{\rm ht}(I(G)),\\ |\text{\rm Minh}(I(G))|&\text{ if }m+\text{\rm ht}(I(H))>n+\text{\rm ht}(I(G)),\\ |\text{\rm Minh}(I(H))|&\text{ if }m+\text{\rm ht}(I(H))<n+\text{\rm ht}(I(G)).\end{array}\right.$$

$$0⟶\frac{S}{I(G):x_1}\stackrel{\cdot x_1}{⟶}\frac{S}{I(G)}⟶\frac{S}{I(G)+⟨x_1⟩}⟶0.$$

$${\beta }_{i,j}(S/I)=\sum _{A\subset [n],|A|=j}{\text{dim}}_{\mathsf{k}}{\tilde{H}}_{j-i-1}({\Delta }_A,\mathsf{k}).$$

$$0\to \frac{S}{I(P_{2n+1}):x_{2n+1}}\to \frac{S}{I(P_{2n+1})}\to \frac{S}{I(P_{2n})+⟨x_{2n+1}⟩}\to 0.$$

$$H\left(\frac{S_{K_{1,s}}}{I(K_{1,s})},t\right)=\frac{1}{(1-t{)}^s}+\frac{1}{(1-t)}-1,$$

$$H\left(\frac{S}{I(G)},t\right)=\frac{e}{1-t}+\frac{e}{(1-t{)}^s}-(2e-1).$$

Full text

1 The author is funded by National Board for Higher Mathematics, India

2 The author is funded by University Grants Commission, India

1. Introduction

Let $G$ be a simple graph on the vertex set $V(G)=\{x_{1},\dots,x_{n}\}$ and edge set $E(G)$. Set $S={\sf k}[x_{1},\dots,x_{n}]$, where ${\sf k}$ is an arbitrary field. The edge ideal of $G$, denoted by $I(G)$, is defined as $I(G):=(x_{i}x_{j}:\{x_{i},x_{j}\}\in E(G))$. From the past one decade, many authors established connections between the combinatorial properties of $G$ and the algebraic properties of $I(G)$. The $s$-th symbolic power of $I(G)$ is defined as follows:

[TABLE]

Let $I$ be a homogeneous ideal in $S$. The regularity of $I$, denoted by ${\rm reg}(I)$, is defined as ${\rm reg}(I):=\max\{j:\operatorname{Tor}_{i}^{S}(I,{\sf k})_{i+j}\neq 0,\;i\geq 0\}$. It is known that ${\rm reg}(S/I)={\rm reg}(I)-1$. Minh stated the following conjecture for the regularity of symbolic power of edge ideal, see [5]:

** Conjecture 1.1****.**

Let $G$ be a finite simple graph and $I(G)$ be its edge ideal. Then,

[TABLE]

Gu et al. studied Conjecture 1.1 for odd cycles in [5]. In [11], Jayanthan and Kumar settled Conjecture 1.1 for graphs, which are clique sum of an odd cycle and some bipartite graphs. In [16] and [17], Fakhari studied Conjecture 1.1 for chordal graphs and unicyclic graphs, respectively. In [15], Selvaraja studied the regularity of $I(G)^{s}$ when $G$ is join of graphs. In this article, we study the regularity of $I(G)^{(s)}$, where $G$ is join of graphs. As a consequence, we prove Conjecture 1.1 for some classes of graphs, for example, wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs.

In [9], Hibi et al. studied the relationship between regularity, degree of $h$-polynomial, and the number of vertices. For a given pair of positive integers $(r,s)$, they constructed a graph whose edge ideal has regularity $r$ and degree of $h$-polynomial $s$. In [8], Hibi et al. constructed a graph for a given pair $(b,r)$ with $b\leq r$, where $b$ is the number of extremal Betti numbers of $S/I(G)$ and $r={\rm reg}(S/I(G))$. In [3], Ramos and Gimenez characterized bipartite graphs whose edge ideals have regularity $3$. In this article, we obtain a class of graphs whose edge ideals have regularity three. Moreover, for a given pair of positive integers $(r,d)$ with $r\leq d$, we construct a graph $G$ such that ${\rm reg}(S/I(G))=r$ and $\text{\rm dim}(S/I(G))=d$. Also, we prove that the multiplicity of edge ideal of a graph has no relation with dimension, depth, regularity, and degree of $h$-polynomial. It is known that the depth and degree of $h$-polynomial for a square-free monomial ideal are bounded above by the dimension. We show that the depth of edge ideal of a graph is independent from the degree of $h$-polynomial and regularity.

Acknowledgement: The authors are grateful to A. V. Jayanthan for his constant support and valuable comments. We want to express our sincere thanks to the referees for a careful reading of the manuscript and raising crucial points.

2. Preliminaries

In this section, we collect all the notation and definitions which are used throughout this article. For the undefined terminology, we refer the readers to the book [7] by Herzog and Hibi.

2.1. Combinatorics

Let $G$ be a simple graph on the vertex set $V(G)$ and edge set $E(G)$.

1. a)

A graph $H$ is called an induced subgraph of $G$ if $V(H)\subset V(G)$ and for $i,j\in V(H)$, $\{i,j\}\in E(H)$ if and only if $\{i,j\}\in E(G)$. 2. b)

The complement of a graph $G$, denoted by $G^{c}$, is the graph on the vertex set $V(G)$ and edge set $\{\{i,j\}:\{i,j\}\notin E(G)\text{ and }i\neq j\}$. 3. c)

A graph $G$ is chordal if there is no induced cycle of length $\geq 4$ and it is co-chordal if $G^{c}$ is chordal. 4. d)

A collection $C\subset E(G)$ of disjoint edges is said to be an induced matching if the induced subgraph on the vertices of $C$ has no edge other than the edges in $C$. The maximum size of an induced matching in $G$ is called the induced matching number of $G$ and is denoted by $\nu(G)$. 5. e)

A collection $\Delta$ of subsets of $[n]$ is called a simplicial complex if it satisfies the following:

1. i)

for any $i\in[n]$, $\{i\}\in\Delta$, 2. ii)

$F\subseteq G$ with $G\in\Delta$ implies that $F\in\Delta$. 6. f)

An element of $\Delta$ is called a face and a maximal face with respect to inclusion is called a facet. 7. g)

The dimension of a simplicial complex $\Delta$ is defined as

[TABLE] 8. h)

Let $G$ and $H$ be graphs on vertex sets $V(G)=\{x_{1},\dots,x_{m}\}$ and $V(H)=\{y_{1},\dots,y_{n}\}$, respectively. Then, the join of $G$ and $H$, denoted by $G\ast H$, is a graph on the vertex set $V(G)\sqcup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{x_{i},y_{j}\}:1\leq i\leq m,1\leq j\leq n\}$. The graph shown in Figure 1 is join of a vertex and cycle $C_{5}$. For a graph $G$ and $l\geq 1$, we denote $G^{\ast l}$ to be join of $l$-copies of $G$. In particular, $G^{\ast 1}=G$.

2.2. Hilbert Series

Let $S={\sf k}[x_{1},\dots,x_{n}]$ be a standard graded polynomial ring and $I\subseteq S$ be a homogeneous ideal with $\text{\rm dim}(S/I)=d$. By [7, Theorem 6.1.3], the Hilbert series of $S/I$ is $H(S/I,t)=(h_{0}+h_{1}t+\dots+h_{s}t^{s})/(1-t)^{d}$ for $h_{i}\in{\mathbb{Z}}$ with $h_{s}\neq 0$.

1. a)

The polynomial $h_{0}+h_{1}t+\dots+h_{s}t^{s}$, denoted by $h_{S/I}(t)$, is called the $h$-polynomial of $S/I$. 2. b)

The multiplicity of $S/I$, denoted by $e(S/I)$, is $h_{S/I}(1)$.

3. Regularity of symbolic powers of join of graphs

In this section, we study the regularity of symbolic powers of join of graphs. First, we prove a technical lemma, which is useful to prove our main result of this section.

** Lemma 3.1****.**

Let $S={\sf k}[x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}]$ be a polynomial ring. Let $I\subset\langle x_{1},\dots,x_{m}\rangle^{2}$ be a square-free monomial ideal in ${\sf k}[x_{1},\ldots,x_{m}]$, and $J\subset\langle y_{1},\dots,y_{n}\rangle^{2}$ be a square-free monomial ideal in ${\sf k}[y_{1},\ldots,y_{n}]$. Then,

[TABLE]

Proof.

Let $K=(I+\langle y_{1},\dots,y_{n}\rangle)^{(s)}+(J+\langle x_{1},\dots,x_{m}\rangle)^{(s)}$. Since $x_{i}^{s},y_{j}^{s}\in K$ for all $i,j$, $K$ is an $\mathfrak{m}$-primary ideal, where $\mathfrak{m}=\langle x_{1},\ldots,x_{m},y_{1},\ldots,y_{n}\rangle$. Therefore,

[TABLE]

It is easy to see that $(S/K)_{j}=0$, for all $j\geq 2s-1$, we have ${\rm reg}(K)\leq 2s-1$. Since $I$ and $J$ are square-free monomial ideals, $x_{1}^{s-1}y_{1}^{s-1}\notin K$. Therefore, $x_{1}^{s-1}y_{1}^{s-1}\in(S/K)_{2s-2}$ is a nonzero element, and the assertion follows. ∎

We now fix some notation which are used through out this section. For $1\leq j\leq r$, set $X_{j}=\{x_{j1},\ldots,x_{jm_{j}}\}$. Let $G_{1},\ldots,G_{r}$ be graphs on vertex sets $X_{1},\ldots,X_{r}$, respectively. Also, for $1\leq j\leq r$, set $\mathfrak{m}_{j}=\langle x_{j1},\ldots,x_{jm_{j}}\rangle$.

** Theorem 3.2****.**

For $r\geq 2$, let $G_{1},\dots,G_{r}$ be graphs on vertex sets $X_{1},\dots,X_{r}$, respectively. Let $G=G_{1}\ast\dots\ast G_{r}$ and $S={\sf k}[V(G)]$. Then,

[TABLE]

Proof.

We prove the assertion by induction on $r$. If $r=2$, then consider the following short exact sequence:

[TABLE]

It follows from [6, Proposition 4.10] that

[TABLE]

Similarly,

[TABLE]

By virtue of Lemma 3.1,

[TABLE]

and hence, it follows from the regularity behavior on short exact sequence that

[TABLE]

Now, the assertion follows from [6, Proposition 4.10]. Assume that $r>2$, and let $G^{\prime}=G_{1}*\cdots*G_{r-1}$. Note that $G=G^{\prime}*G_{r}$. Hence, the result follows from induction hypothesis and the base case. ∎

In [15], Selvaraja defined following classes of graphs:

[TABLE]

where $\mathcal{G}(I(G)^{s})$ denotes the minimal monomial generating set of $I(G)^{s}$, and

[TABLE]

It follows from [15, Theorem 4.4, Remark 4.10] that $\mathcal{A}$ as well as $\mathcal{A}\cap\mathcal{A}_{1}$ are closed under the operation of join of graphs. We study the regularity of symbolic power of join of graphs which belong to class $\mathcal{A}\cap\mathcal{A}_{1}$. There are many classes of graphs which are contained in $\mathcal{A}\cap\mathcal{A}_{1}$. For example, co-chordal graphs, unmixed bipartite, bipartite $P_{6}$-free graphs, forest graphs are contained in $\mathcal{A}\cap\mathcal{A}_{1}$, for more details see [15]. It follows from [12, Corollary 4.6] that if $G$ is a chordal graph, then $G\in\mathcal{A}_{1}$. Also, it follows from the proof of [12, Theorems 4.1, 4.4] that $G\in\mathcal{A}$ if $G$ is a chordal graph.

** Corollary 3.3****.**

Let $G_{1},\dots,G_{r}\in\mathcal{A}_{1}$ such that $G_{i}$’s are bipartite or chordal and $G=G_{1}\ast\dots\ast G_{r}$. Then, ${\rm reg}\left(I(G)^{(s)}\right)=2s+{\rm reg}(I(G))-2,\text{ for all }s\geq 1.$ Furthermore, if $G_{i}\in\mathcal{A}$ for all $i$, then ${\rm reg}\left(I(G)^{(s)}\right)={\rm reg}(I(G)^{s}),\text{ for all }s\geq 1$. In particular, ${\rm reg}(I(G)^{(s)})=2s$ if $G$ is a complete multipartite graph.

Proof.

If $G_{i}$ is bipartite and $G_{i}\in\mathcal{A}_{1}$, then we have

[TABLE]

If $G_{i}$ is a chordal graph, then by [16, Theorem 3.3], ${\rm reg}(I(G_{i})^{(s)})=2s+\nu(G_{i})-1$. Therefore, by virtue of Theorem 3.2, we get ${\rm reg}\left(I(G)^{(s)}\right)=\max\left\{{\rm reg}\left(I(G_{i})^{(s)}\right):1\leq i\leq r\right\}.$ Since for all $1\leq i\leq r$, ${\rm reg}(I(G_{i})^{(s)})=2s+{\rm reg}(I(G_{i}))-2$, it follows from [14, Proposition 3.1.2] that

[TABLE]

Now, assume that each $G_{i}\in\mathcal{A}$. By [15, Remark 4.10], ${\rm reg}\left(I(G)^{(s)}\right)={\rm reg}\left(I(G)^{s}\right)$ which completes the proof. ∎

In [15], Selvaraja studied the regularity of powers of edge ideals of wheel graphs. In the following result, we prove Conjecture 1.1 for wheel graphs.

** Corollary 3.4****.**

Let $G=W_{n}$ be a wheel graph for $n\geq 4$. Then,

[TABLE]

Proof.

Since $G=W_{n}$ is join of a vertex $v$ and a cycle $C_{n}$, by Theorem 3.2, we have ${\rm reg}(I(G)^{(s)})=\max\{{\rm reg}I(C_{n})^{(i)}-i+s:1\leq i\leq s\}.$ It follows from [5, Theorem 5.3] that for $i\geq 2$, ${\rm reg}(I(C_{n})^{(i)})={\rm reg}(I(C_{n})^{i})=2i+\nu(C_{n})-1$, and hence, our result follows from [15, Theorem 4.7]. ∎

4. Construction of Graphs

In this section, we construct graphs whose edge ideals have different pair of algebraic invariants. Let $G$ and $H$ be graphs on vertex sets $V(G)=\{x_{1},\dots,x_{m}\}$ and $V(H)=\{y_{1},\dots,y_{n}\}$, respectively. One can see that if $C$ is a minimal vertex cover of $G*H$, then either $C=A\cup V(H)$ or $C=V(G)\cup B$, where $A$ and $B$ are minimal vertex covers of $G$ and $H$, respectively. Therefore, we have the following short exact sequence

[TABLE]

where $\mathfrak{m}_{1}=\langle x_{1},\dots,x_{m}\rangle$ and $\mathfrak{m}_{2}=\langle y_{1},\dots,y_{n}\rangle$ and $S={\sf k}[x_{1},\dots,x_{m},y_{1},\dots,y_{n}].$ Thus, by depth lemma, $0pt(S/I(G\ast H))=1$.

Note that for a square-free monomial ideal $I$, by [13, Observation 5.2], $e(S/I)=|\text{\rm Minh}(I)|$, where $\text{\rm Minh}(I)=\{\mathfrak{p}\in\text{\rm Ass}(S/I):\text{\rm ht}(\mathfrak{p})=\text{\rm ht}(I)\}$. We now compute the multiplicity of join of graphs.

** Theorem 4.1****.**

Let $G$ and $H$ be graphs on vertex sets $V(G)=\{x_{1},\ldots,x_{m}\}$ and $V(H)=\{y_{1},\ldots,y_{n}\}$, respectively. Suppose $S={\sf k}[x_{1},\dots,x_{m},y_{1},\dots,y_{n}]$. Then,

[TABLE]

Proof.

For a graph $G$, it is well known that minimal primes of $I(G)$ correspond to minimal vertex covers of $G$. Note that if $\mathfrak{p}\in\text{\rm Minh}(I(G\ast H))$, then either $\mathfrak{p}=\mathfrak{p}_{1}+\mathfrak{m}_{2}$ or $\mathfrak{p}=\mathfrak{p}_{2}+\mathfrak{m}_{1}$, where $\mathfrak{p}_{1}\in\text{\rm Minh}(I(G))$ and $\mathfrak{p}_{2}\in\text{\rm Minh}(I(H))$. Thus, we get

[TABLE]

Hence, the assertion follows. ∎

Now, we prove the algebraic properties of join of graphs which are used in rest of the section.

** Proposition 4.2****.**

Let $G^{*l}=G\ast\dots\ast G$ be join of $l$-copies of $G$, $S_{G}={\sf k}[V(G)]$ and $S={\sf k}[V(G^{*l})].$ Then,

1. a)

${\rm reg}(S/I(G^{*l}))={\rm reg}(S_{G}/I(G))$, 2. b)

$e(S/I(G^{*l}))=l\cdot e(S_{G}/I(G))$, 3. c)

$H(S/I(G^{*l}),t)=l\cdot H(S_{G}/I(G),t)-(l-1)$.

Proof.

(a) and (c) follow from [14, Proposition 3.12, Corollary 4.6], respectively. (b) can be obtained by recursively applying Theorem 4.1. ∎

Using above result, one can see that $e(S/I(K_{n}))=n$, and using (1), $0pt(S/I(K_{n}))=1$.

** Notation 4.3****.**

Let $K_{n}$ be the complete graph on the vertex set $\{x_{1},\dots,x_{n}\}$. Then, for $1\leq r\leq n$, we define a graph $W(n,r)$ on the vertex set $\{x_{1},\dots,x_{n},y_{1},\dots,y_{r}\}$ with edge set $E(K_{n})\cup\{\{x_{i},y_{i}\}:1\leq i\leq r\}$. The graph shown in Figure 2 is $W(5,3)$.

Now, we study the algebraic invariants of $W(n,r)$.

** Theorem 4.4****.**

Let $G=W(n,r)$ and $S={\sf k}[V(G)]$. Then, we have the following:

1. a)

$I(G)$* has linear resolution,* 2. b)

$\text{\rm pd}(S/I(G))=n$* and $0pt(S/I(G))=r$,* 3. c)

if $1\leq r<n$, then $e(S/I(G))=n-r$.

Proof.

(a) It is easy to see that $W(n,r)$ is a co-chordal graph. Hence, the assertion follows from [4, Theorem 1].

(b) We proceed by induction on $n$. Consider the following short exact sequence

[TABLE]

Note that $I(G):x_{1}=\langle x_{2},\ldots,x_{n},y_{1}\rangle$ and $I(G)+\langle x_{1}\rangle=I(W(n-1,r-1))+\langle x_{1}\rangle$. Since $I(G):x_{1}$ is generated by a regular sequence of length $n$, $\text{\rm pd}(S/(I(G):x_{1}))=n$ and $0pt(S/(I(G):x_{1}))=r$. Observe that $y_{1}$ is a regular element on $S/(I(G)+\langle x_{1}\rangle)$. Thus, by induction, $0pt(S/(I(G)+\langle x_{1}\rangle))=r$. Now, by applying depth lemma on the above short exact sequence, we have $0pt(S/I(G))=r$.

(c) We use induction on $n$. Let $A$ be a minimal vertex cover of $G$. Then, $|A\cap\{x_{1},\ldots,x_{n}\}|\geq n-1$. Also, $\{x_{1},\ldots,x_{n-1}\}$ is a minimal vertex cover of $G$. Therefore, $\text{\rm ht}(I(G))=n-1$, and hence, $\text{\rm dim}(S/I(G))=r+1$. From the proof of (b), we have $\text{\rm dim}(S/(I(G):x_{1}))=r$ and $\text{\rm dim}(S/(I(G)+\langle x_{1}\rangle))=r+1$. Thus, by the above short exact sequence, $e(S/I(G))=e(S/(I(G)+\langle x_{1}\rangle))$. Hence, the assertion follows by induction. ∎

In the following remark, we prove that ${\rm reg}(S/I)\leq\text{\rm dim}(S/I)$, if $I$ is a square-free monomial ideal.

** Remark 4.5****.**

It is a known result, but for the sake of completeness, we prove it. Let $I\subset S={\sf k}[x_{1},\dots,x_{n}]$ be a square-free monomial ideal. Let $\text{\rm dim}(S/I)=d$. Since $I$ is a square-free monomial ideal, there exists a simplicial complex $\Delta$ on $[n]$ such that the Stanley-Reisner ideal of $\Delta$ is $I$. It follows from [7, Section 1.5] that $\text{\rm dim}(\Delta)=d-1$. Therefore, $\tilde{H}_{l}(\Delta,{\sf k})=0$, for $l\geq d$. By Hochster’s formula [7, Theorem 8.1.1],

[TABLE]

Let ${\rm reg}(S/I)=r$. Then, there exists $i,j$ such that $j-i=r$ and $\beta_{i,j}(S/I)\neq 0$. Therefore, for some subset $A\subset[n]$, $\text{\rm dim}_{{\sf k}}\tilde{H}_{r-1}(\Delta_{A},{\sf k})\neq 0$, which implies that $r-1\leq d-1$, and hence, ${\rm reg}(S/I)\leq\text{\rm dim}(S/I)$.

As a consequence, we have the following:

** Corollary 4.6****.**

Let $G$ be a triangle free graph which is not a forest. Then, ${\rm reg}(S/I(G^{c}))=2$.

Proof.

It is a well known fact that $\text{\rm dim}(S/I(G^{c}))$ is the maximum size of a maximal independent set of $G^{c}$ which is equal to the maximum size of a maximal clique in $G$. Since $G$ is triangle free, the maximum size of any maximal clique is $2$. Hence, by Remark 4.5, ${\rm reg}(S/I(G^{c}))\leq 2$. Further, by [4, Theorem 1], ${\rm reg}(S/I(G^{c}))=2$ as $G$ is not chordal. ∎

** Remark 4.7****.**

Let $G$ and $H$ be graphs with ${\rm reg}(I(G))=3$ and ${\rm reg}(I(H))\leq 3$. Then, by [14, Proposition 3.12], ${\rm reg}(I(G\ast H))=3$. For example, if $G^{c}$ is triangle free which is not a forest and $H$ is a bipartite graph whose edge ideal has regularity $3$ ([3, Theorem 4.1]) or co-chordal graph, then by Corollary 4.6, ${\rm reg}(I(G\ast H))=3$. In this way, one can construct graphs whose edge ideal have regularity $3$ which are neither bipartite graphs nor compliment of triangle free graphs.

Let $F_{n}$ denote the bipartite graph on the vertex set $\{x_{1},\dots,x_{n},y_{1},\dots,y_{n}\}$ with edge set $\{\{x_{i},y_{j}\}:1\leq i\leq j\leq n\}$. By virtue of [7, Corollarly 9.1.14], $S/I(F_{n})$ is Cohen-Macaulay. Since $\{x_{1},\ldots,x_{n}\}$ is a minimal vertex cover of $F_{n}$, $\text{\rm ht}(I(F_{n}))=\text{\rm dim}(S/I(F_{n}))=n$. Observe that $\mu(I(F_{n}))=\binom{n+1}{2}$. Thus, it follows from [18, Theorem 4.3.7] that $I(F_{n})$ has linear resolution. The graph shown in Figure 3 is $F_{4}$.

Now, for $1\leq r\leq d$, we construct a graph $G$ with ${\rm reg}(S/I(G))=r$ and $\text{\rm dim}(S/I(G))=d$.

** Theorem 4.8****.**

For $1\leq r\leq d$, there exists a graph $G$ such that ${\rm reg}(S/I(G))=r$ and $\text{\rm dim}(S/I(G))=d$.

Proof.

Let $G$ be a disconnected graph with $r$ components such that the $r-1$ components are edges and one component is $F_{d-r+1}$. Then, ${\rm reg}(S/I(G))=r$ and $\text{\rm dim}(S/I(G))=d$. ∎

Note that $\{x_{2i}:1\leq i\leq n\}$ is a minimal vertex cover of both $P_{2n}$ and $P_{2n+1}$. Therefore, $\text{\rm ht}(I(P_{2n}))\leq n$ and $\text{\rm ht}(I(P_{2n+1}))\leq n$. Now, if possible, let $A$ be a minimal vertex cover of $P_{2n}$ such that $|A|<n$. Then, it is easy to see that $|\{e:e\in E(P_{2n})\text{ and }e\cap A\neq\emptyset\}|\leq 2n-2$, which is a contradiction. Similarly, one can prove that $\text{\rm ht}(I(P_{2n+1}))=n$. Hence, $\text{\rm dim}(S/I(P_{2n}))=n$ and $\text{\rm dim}(S/I(P_{2n+1}))=n+1$.

** Lemma 4.9****.**

Let $G=P_{2n+1}$ be a path graph. Then, $e(S/I(P_{2n+1}))=1$.

Proof.

We prove the assertion by induction on $n$. If $n=1$, then $P_{3}$ has only one minimal vertex cover of minimal size. Hence, $e(S/I(P_{3}))=1$. Assume that $n>1$, and $e(S/I(P_{2n-1}))=1$. Now, consider the following short exact sequence,

[TABLE]

Note that $I(P_{2n+1}):x_{2n+1}=I(P_{2n-1})+\langle x_{2n}\rangle$. Observe that $\text{\rm dim}(S/(I(P_{2n})+\langle x_{2n+1}\rangle))=n$, $\text{\rm dim}(S/(I(P_{2n+1}):x_{2n+1}))=n+1$ and $\text{\rm dim}(S/I(P_{2n+1}))=n+1$. Hence, $e(S/I(P_{2n+1}))=e(S/(I(P_{2n-1})+\langle x_{2n}\rangle))=1$. ∎

** Remark 4.10****.**

Let $I\subseteq\mathfrak{m}^{2}$ be a nonzero homogeneous ideal in a polynomial ring $S$ with $H(S/I,t)=(1+h_{1}t+h_{2}t^{2}+\dots+h_{s}t^{s})/(1-t)^{d}$, where $d=\text{\rm dim}(S/I)$. Then, $h_{1}=\text{\rm codim}(S/I)$. This implies that for any graph $G$ if $e(S/I(G))=1$, then $\deg(h_{S/I(G)}(t))\geq 2$. Note that if $I$ is a square-free monomial ideal, then by [2, Theorem 5.1.7], $\deg(h_{S/I}(t))\leq\text{\rm dim}(S/I)$. Therefore, $e(S/I(G))=1$ implies that $\text{\rm dim}(S/I(G))\geq 2$.

We now prove that the multiplicity of edge ideal of a graph is not bounded by algebraic invariants such as regularity, depth, dimension, and degree of $h$-polynomial. We construct graphs with fixed multiplicity and one of the other invariants.

** Theorem 4.11****.**

Let $e,r,s,\delta,d$ be positive integers. Then, we have the following:

1. a)

There exists a graph $G$ with $e(S/I(G))=e$ and ${\rm reg}(S/I(G))=r$. 2. b)

There exists a graph $G$ with $e(S/I(G))=e$ and $\deg(h_{S/I(G)}(t))=s$, provided $e\cdot s\geq 2$. 3. c)

There exists a graph $G$ with $e(S/I(G))=e$ and $0pt(S/I(G))=\delta$. 4. d)

There exists a graph $G$ with $e(S/I(G))=e$ and $\text{\rm dim}(S/I(G))=d$, provided $e\cdot d\geq 2$.

Observe that by Proposition 4.2, the regularity and dimension remain the same under self join operation but multiplicity increases. This observation is the key idea for the proof of Theorem 4.11.

Proof of Theorem 4.11.

(a) If $r$ is odd, then take $H=P_{3r}$, otherwise take $H=P_{3r+1}$. Then, by [1, Theorem 4.7], ${\rm reg}(S_{H}/I(H))=r$ and by Lemma 4.9, $e(S_{H}/I(H))=1$, where $S_{H}={\sf k}[V(H)]$. Now, take $G=H^{\ast e}$ and $S={\sf k}[V(G)]$. Then, by Proposition 4.2, we have ${\rm reg}(S/I(G))=r$ and $e(S/I(G))=e$.

(b) and (d): For $d=s=1$, take $G=K_{e}$. Now, we assume that $s\geq 2$, and $d\geq 2$. By short exact sequence (1), we get

[TABLE]

where $S_{K_{1,s}}={\sf k}[V({K_{1,s}})].$ Take $G=K_{1,s}^{\ast e}$. Using Proposition 4.2, we have

[TABLE]

Thus, $e(S/I(G))=e$ and $\deg(h_{S/I(G)}(t))=s=\text{\rm dim}(S/I(G))$.

(c) For any $e$ and $\delta$, take $G=W(e+\delta,\delta)$. Then, it follows from Theorem 4.4 that $e(S/I(G))=e$ and $0pt(S/I(G))=\delta$. ∎

We now prove that the depth of edge ideal of a graph is independent from regularity and degree of $h$-polynomial. Indeed, we construct graph whose edge ideal has depth $\delta$ and regularity $r$. Also, we show the existence of a graph whose edge ideal has depth $\delta$ and degree of $h$-polynomial $s$.

** Theorem 4.12****.**

Let $\delta,r,s$ be positive integers. Then, we have the following:

1. a)

There exists a graph $G$ such that $0pt(S/I(G))=\delta$ and ${\rm reg}(S/I(G))=r$. 2. b)

There exists a graph $G$ such that $0pt(S/I(G))=\delta$ and $\deg(h_{S/I(G)}(t))=s$.

First, we prove the above theorem for $\delta=1$ or $r=1$ or $s=1$. Then, we use the following fact to complete our theorem: Let $I\subset R={\sf k}[x_{1},\dots,x_{m}]$ and $J\subset T={\sf k}[y_{1},\dots,y_{n}]$ be homogeneous ideals and $S={\sf k}[x_{1},\dots,x_{m},y_{1},\dots,y_{n}]$. Then, it follows from [10, Lemma 1,5] that $\deg(h_{S/(I+J)}(t))=\deg(h_{R/I}(t))+\deg(h_{T/J}(t))$. By [10, Lemma 2.2], $0pt(S/(I+J))=0pt(R/I)+0pt(T/J)$ and by [10, Lemma 3.2], ${\rm reg}(S/(I+J))={\rm reg}(R/I)+{\rm reg}(T/J)$.

Proof of Theorem 4.12.

(a) Case I. For $\delta=1$ and $r\geq 1$, take $G=P_{3r}^{\ast 2}$. By Proposition 4.2 and [1, Theorem 4.7], ${\rm reg}(S/I(G))=r$, and by the short exact sequence (1), $0pt(S/I(G))=1$.

Case II. For $2\leq\delta\leq r$, take $G$ to be the union of $P_{3(r-\delta+1)}^{\ast 2}$ and $\delta-1$ disjoint edges. Then, we get $0pt(S/I(G))=\delta$ and ${\rm reg}(S/I(G))=r$.

Case III. For $r=1$, consider $G=W(\delta+1,\delta)$. By Theorem 4.4, ${\rm reg}(S/I(G))=1$ and $0pt(S/I(G))=\delta$.

Case IV. For $2\leq r<\delta$, take $G$ to be the union of $W(\delta-r+2,\delta-r+1)$ and $r-1$ disjoint edges. This completes the proof of (a).

(b) Case I. For $\delta=1$, consider $G=K_{1,s}$. It follows from (1) that $0pt(S/I(G))=1$ and by (2), $\deg(h_{S/I(G)}(t))=s$.

Case II. For $2\leq\delta\leq s$, take $G$ to be the union of $K_{1,s-\delta+1}$ and $\delta-1$ disjoint edges. Then, $0pt(S/I(G))=\delta$ and $\deg(h_{S/I(G)}(t))=s$.

Case III. For $s=1$, take $G=F_{\delta}$. Since $S/I(G)$ is Cohen-Macaulay of dimension $\delta$ and ${\rm reg}(S/I(G))=1$, we have $0pt(S/I(G))=\delta$ and $\deg(h_{S/I(G)}(t))=1$.

Case IV. For $2\leq s<\delta$, take $G$ to be the union of $F_{\delta-s+1}$ and $s-1$ disjoint edges. Now, the assertion follows from Case III. ∎