Regularity and $a$-invariant of Cameron--Walker graphs

TL;DR

This paper investigates the algebraic properties of Cameron--Walker graphs, showing that their associated edge ideals have a specific invariant value and satisfy a key equality relating regularity, dimension, and depth.

Contribution

It establishes that all Cameron--Walker graphs have an $a$-invariant of zero and identifies conditions under which their edge ideals satisfy a fundamental algebraic equality.

Findings

All Cameron--Walker graphs have $a$-invariant equal to zero.

A class of Cameron--Walker graphs satisfies $s - r = d - e$.

The paper links graph structure to algebraic invariants of their edge ideals.

Abstract

Let $S$ be the polynomial ring over a field $K$ and $I \subset S$ a homogeneous ideal. Let $h(S/I,\lambda)$ be the $h$-polynomial of $S/I$ and $s = \mathrm{deg} h(S/I,\lambda)$ the degree of $h(S/I,\lambda)$. It follows that the inequality $s - r \leq d - e$, where $r = \mathrm{reg} (S/I)$, $d = \dim S/I$ and $e = \mathrm{depth} S/I$, is satisfied and, in addition, the equality $s - r = d - e$ holds if and only if $S/I$ has a unique extremal Betti number. We are interested in finding a natural class of finite simple graphs $G$ for which $S/I(G)$, where $I(G)$ is the edge ideal of $G$, satisfies $s - r = d - e$. Let $a(S/I(G))$ denote the $a$-invariant of $S/I$, i.e., $a(S/I(G)) = s - d$. One has $a(S/I(G)) \leq 0$. In the present paper, by showing the fundamental fact that every Cameron--Walker graph $G$ satisfies $a(S/I(G)) = 0$, a class of Cameron--Walker graphs $G$ for which $S/I(G)$ satisfies $s - r = d - e$ will be exhibited.

Equations

$$H\left(S/I,\lambda \right)=\frac{h_0+h_1\lambda +h_2{\lambda }^2+\cdots +h_s{\lambda }^s}{(1-\lambda {)}^d},$$

$$h\left(S/I,\lambda \right)=h_0+h_1\lambda +h_2{\lambda }^2+\cdots +h_s{\lambda }^s$$

$${\mathbf{F}}_{S/I}:0\to \underset{j\ge 1}{⨁}S(-(p+j){)}^{{\beta }_{p,p+j}(S/I)}\to \cdots \to \underset{j\ge 1}{⨁}S(-(1+j){)}^{{\beta }_{1,1+j}(S/I)}\to S\to S/I\to 0$$

$$\mathrm{reg}\left(S/I\right)=\max \{j:{\beta }_{i,i+j}(S/I)\ne 0\}.$$

$$\mathrm{deg}h\left(S/I,\lambda \right)-\mathrm{reg}\left(S/I\right)\le \dim S/I-\mathrm{depth}\left(S/I\right)$$

$$H\left(S/I,\lambda \right)=\frac{{\sum }_{i=0}^p(-1{)}^i{\sum }_{j\in \mathbb{Z}}{\beta }_{i,i+j}(S/I){\lambda }^{i+j}}{(1-\lambda {)}^n}=\frac{h(S/I,\lambda )\cdot (1-\lambda {)}^{n-\dim S/I}}{(1-\lambda {)}^n},$$

$$(\ast )\mathrm{deg}h\left(S/I,\lambda \right)-\mathrm{reg}\left(S/I\right)=\dim S/I-\mathrm{depth}\left(S/I\right)$$

$$I(G)=\left(x_i{x}_j:\{x_i,x_j\}\in E(G)\right)\subset S.$$

$$im(G)\le \mathrm{reg}\left(S/I(G)\right)\le m(G)$$

$$V(G)=\bigcup _{i=1}^m\left\{x_1^{(i)},\dots ,x_{s_i}^{(i)}\right\}\cup \{v_1,\dots ,v_m\}\cup \{w_1,\dots ,w_n\}\cup \left\{\bigcup _{j=1}^n\bigcup _{\ell =1}^{t_j}\left\{y_{\ell ,1}^{(j)},y_{\ell ,2}^{(j)}\right\}\right\},$$

$$\mathrm{deg}h\left(K[V(G)]/I(G),\lambda \right)=\dim K[V(G)]/I(G)=\sum _{i=1}^m{s}_i+\sum _{j=1}^n\max \left\{t_j,1\right\}.$$

$$H\left(S/I,\lambda \right)=H\left(S/I+(x),\lambda \right)+\lambda \cdot H\left(S/I:(x),\lambda \right).$$

$$\frac{K[V(G)]}{I(G)+(x_v)}\cong \frac{K[V(G)\setminus \{x_v\}]}{I\left(G_{V(G)\setminus \{x_v\}}\right)},$$

$$\frac{K[V(G)]}{I(G):(x_v)}\cong \frac{K[V(G)\setminus N_G[x_v]]}{I\left(G_{V(G)\setminus N_G[x_v]}\right)}{\otimes }_{K}K[x_v].$$

$$H\left(K[V(G)]/I(G),\lambda \right)$$

$$H\left(K[V(G_s^{\mathrm{star}(x_v)})]/I(G_s^{\mathrm{star}(x_v)}),\lambda \right)=\frac{1+\lambda (1-\lambda {)}^{s-1}}{(1-\lambda {)}^s}.$$

$$\mathrm{deg}h\left(K[V(G_s^{\mathrm{star}(x_v)})]/I(G_s^{\mathrm{star}(x_v)}),\lambda \right)=\dim K[V(G_s^{\mathrm{star}(x_v)})]/I(G_s^{\mathrm{star}(x_v)})=s.$$

$$H\left(K[V(G_t^{△(x_v)})]/I(G_t^{△(x_v)}),\lambda \right)=\frac{(1+\lambda {)}^t+\lambda (1-\lambda {)}^{t-1}}{(1-\lambda {)}^t}.$$

$$\mathrm{deg}h\left(K[V(G_t^{△(x_v)})]/I(G_t^{△(x_v)}),\lambda \right)=\{\begin{array}{ll}t& \text{(t : odd)}\\ t-1& \text{(t : even)}\end{array}$$

$$H\left(S/I_1+I_2,\lambda \right)=H\left(S_1/I_1,\lambda \right)\cdot H\left(S_2/I_2,\lambda \right).$$

$$\mathrm{deg}h\left(S/I_1+I_2,\lambda \right)=\mathrm{deg}h\left(S_1/I_1,\lambda \right)+\mathrm{deg}h\left(S_2/I_2,\lambda \right),$$

$$\dim S/I_1+I_2=\dim S_1/I_1+\dim S_2/I_2.$$

$$\mathrm{deg}h\left(K[V(G)]/I(G),\lambda \right)=\sum _{i=1}^r\mathrm{deg}h\left(K[V(G_i]/I(G_i),\lambda \right),$$

$$\dim K[V(G)]/I(G)=\sum _{i=1}^r\dim K[V(G_i)]/I(G_i),$$

$$\mathrm{deg}h\left(K[V(G)]/I(G),\lambda \right)=\dim K[V(G)]/I(G)=s_1+t_1.$$

$$H\left(\frac{K[V(G)\setminus \{v_1\}]}{I\left(G_{V(G)\setminus \{v_1\}}\right)},\lambda \right)=\frac{(1+\lambda {)}^{t_1}+\lambda (1-\lambda {)}^{t_1-1}}{(1-\lambda {)}^{s_1+t_1}}$$

$$H\left(\frac{K[V(G)\setminus N_G[v_1]]}{I\left(G_{V(G)\setminus N_G[v_1]}\right)},\lambda \right)=\frac{(1+\lambda {)}^{t_1}}{(1-\lambda {)}^{t_1}}.$$

$$H(K[V(G)]/I(G),\lambda )$$

$$\mathrm{deg}h\left(K[V(G)]/I(G),\lambda \right)=\dim K[V(G)]/I(G)=s_1+\sum _{j=1}^{n-1}\max \{t_j,1\}+1.$$

$$\mathrm{deg}h\left(K[V(G)]/I(G),\lambda \right)=\dim K[V(G)]/I(G)=s_1+1+\sum _{j=1}^{n-1}\max \{t_j,1\},$$

Full text

Introduction

In the current trends on combinatorial and computational commutative algebra, the study on regularity of edge ideals of finite simple graphs becomes fashionable and many papers including [3, 8, 9, 17, 21] have been published. In the present paper we are interested in the regularity and the $h$-polynomials of edge ideals.

Let $S=K[x_{1},\ldots,x_{n}]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_{i}=1$ and $I\subset S$ a homogeneous ideal of $S$ with $\dim S/I=d$. The Hilbert series $H\left(S/I,\ \lambda\right)$ of $S/I$ is of the form

[TABLE]

where each $h_{i}\in{\mathbb{Z}}$ ([5, Proposition 4.4.1]). We say that

[TABLE]

with $h_{s}\neq 0$ is the $h$-polynomial of $S/I$. We call the difference $\deg h\left(S/I,\ \lambda\right)-\dim S/I$ the $a$-invariant ([5, Definition 4.4.4]) of $S/I$ and denote it by $a(S/I)$. It is known that $a(S/I)\leq 0$ if $I$ is a squarefree monomial ideal.

Let

[TABLE]

be the minimal graded free resolution of $S/I$ over $S$, where $p$ is the projective dimension of $S/I$. The (Castelnuovo–Mumford ) regularity of $S/I$ is

[TABLE]

The inequality

[TABLE]

is well known ([20, Corollary B.4.1]) and its proof is easy. In fact, since [5, Lemma 4.1.13] says that

[TABLE]

it follows that $\deg h\left(S/I,\ \lambda\right)\leq p+\operatorname{reg}\left(S/I\right)-n+\dim S/I$. Furthermore, since $n-p=\operatorname{depth}(S/I)$ by Auslander–Buchsbaum Theorem, the inequality (0.1) follows. In addition, the equality

[TABLE]

holds if and only if $\beta_{p,p+\operatorname{reg}(S/I)}(S/I)\neq 0$, in other words, if and only if $S/I$ has a unique extremal Betti number ([10, Definition 4.3.13]). In particular, the equality $(*)$ holds if $S/I$ is Cohen–Macaulay by [2, Lemma 3] or $I$ has a pure resolution ([5, p. 153]).

Let $G$ be a finite simple graph (i.e. a graph with no loop and no multiple edge) on the vertex set $V(G)=\{x_{1},x_{2},\ldots,x_{n}\}$ and its edge set $E(G)$. Set $S=K[V(G)]$. The edge ideal of $G$ is

[TABLE]

It is natural to ask for which graph $G$, its edge ideal $I(G)$ satisfies $a(S/I(G))=0$ or the equality $(\ast)$. In the present paper we focus on Cameron–Walker graphs. Let us recall the definition of a Cameron–Walker graph. Let $im(G)$ (resp.  $m(G)$) denote the induced matching number (resp.  matching number) of $G$, see [11, p.258]. Then for any finite simple graph $G$, one has

[TABLE]

by virtue of [9, Theorem 6.7] and [15, Lemma 2.2]. Cameron and Walker [6, Theorem 1] (see also [11, Remark 0.1]) characterized a finite connected simple graph $G$ satisfying $im(G)=m(G)$. A Cameron–Walker graph $G$ is a graph satisfying $im(G)=m(G)$ which is neither a star graph nor a star triangle; see Section 1 for more detail. In [11, 19], Cameron–Walker graphs have been studied from a viewpoint of commutative algebra.

In the present paper, we first prove $a(S/I(G))=0$ for every Cameron–Walker graph $G$ (Theorem 1.1) in Section 1. We next give a classification of Cameron–Walker graphs $G$ whose edge ideal $I(G)$ satisfies the equality $(*)$ (Theorem 2.2) in Section 2. We also provide some classes of graphs other than Cameron–Walker graphs satisfying $(\ast)$ (Proposition 2.10). In general, there is no relationship between the degree of the $h$-polynomial and the regularity even for edge ideals; see [13]. However we prove in Section 3 that for a Cameron–Walker graph $G$, the inequality $\deg h\left(S/I(G),\lambda\right)\geq\operatorname{reg}(S/I(G))$ holds. Moreover we characterize the Cameron–Walker graphs $G$ which satisfy the equality (Theorem 3.1).

1. $a$-invariant of Cameron–Walker graphs

In this section, we show

Theorem 1.1**.**

Let $G$ be a Cameron–Walker graph. Then $a(K[V(G)]/I(G))=0$.

We first recall the definition of a Cameron–Walker graph. Let $G$ be a finite simple graph on the vertex set $V(G)$ with the edge set $E(G)$. We call a subset $\mathcal{M}\subset E(G)$ a matching of $G$ if $e\cap e^{\prime}=\emptyset$ for any $e,e^{\prime}\in\mathcal{M}$ with $e\neq e^{\prime}$. A matching $\mathcal{M}$ of $G$ is called an induced matching of $G$ if for $e,e^{\prime}\in\mathcal{M}$ with $e\neq e^{\prime}$, there is no edge $f\in E(G)$ with $e\cap f\neq\emptyset$ and $e^{\prime}\cap f\neq\emptyset$. The matching number $m(G)$ of $G$ is the maximum cardinality of the matchings of $G$. Also the induced matching number $im(G)$ of $G$ is the maximum cardinality of the induced matchings of $G$. As noted in Introduction, the inequalities $im(G)\leq\operatorname{reg}\left(K[V(G)]/I(G)\right)\leq m(G)$ hold. By virtue of [6, Theorem 1] together with [11, Remark 0.1], the equality $im(G)=m(G)$ holds if and only if $G$ is one of the following graphs:

• •

a star graph, i.e. a graph joining some paths of length $1$ at one common vertex (see Figure 2);

• •

a star triangle, i.e. a graph joining some triangles at one common vertex (see Figure 3);

• •

a connected finite graph consisting of a connected bipartite graph with vertex partition $\{v_{1},\ldots,v_{m}\}\cup\{w_{1},\ldots,w_{n}\}$ such that there is at least one leaf edge attached to each vertex $v_{i}$ and that there may be possibly some pendant triangles attached to each vertex $w_{j}$. Here a leaf edge is an edge meeting a vertex of degree $1$ and a pendant triangle is a triangle whose two vertices have degree $2$ and the rest vertex has degree more than $2$.

We say that a finite connected simple graph $G$ is Cameron–Walker if $im(G)=m(G)$ and if $G$ is neither a star graph nor a star triangle.

Remark 1.2**.**

One can consider a star graph $G$ with $|V(G)|\geq 3$ as a Cameron–Walker graph consisting of bipartite graph $\mathcal{K}_{1,1}$ with some leaf edges and without pendant triangle. Hence claims for Cameron–Walker graph in the below are also true for such a star graph.

Note that for a Cameron–Walker graph $G$, the regularity of $K[V(G)]/I(G)$ is equal to $im(G)$ (equivalently, $m(G)$).

Let $G$ be a Cameron–Walker graph. In what follows we use the following labeling on vertices of $G$; see Figure 1:

[TABLE]

where $\{v_{1},\ldots,v_{m}\}\cup\{w_{1},\ldots,w_{n}\}$ is a vertex partition of a connected bipartite subgraph of $G$, $x_{k}^{(i)}$ ($i=1,\ldots,m$; $k=1,\ldots,s_{i}$) is a vertex such that $\left\{v_{i},x_{k}^{(i)}\right\}$ is a leaf edge, and $y_{\ell,1}^{(j)},y_{\ell,2}^{(j)}$ ($j=1,\ldots,n$; $\ell=1,\ldots,t_{j}$) are vertices which together with $w_{j}$ form a pendant triangle. Note that $s_{i}\geq 1$ and $t_{j}\geq 0$.

We prove Theorem 1.1 by showing

Proposition 1.3**.**

Let $G$ be a Cameron–Walker graph as in Figure 1. Then

[TABLE]

Before giving a proof of Proposition 1.3, several lemmata will be prepared. Let $I\subset S$ be a monomial ideal of $S$ and let $x$ be a variable of $S$ which appears in some monomial belonging to the unique minimal system of monomial generators of $I$. Then, by the additivity of Hilbert series on the exact sequence $0\rightarrow S/I:(x)(-1)\xrightarrow{\ \cdot x\ }S/I\rightarrow S/I+(x)\rightarrow 0$, one has

Lemma 1.4**.**

[TABLE]

Let $G$ be a finite simple graph on the vertex set $V(G)=\{x_{1},\ldots,x_{n}\}$ with the edge set $E(G)$. For $W\subset V(G)$, the induced subgraph $G_{W}$ is the subgraph of $G$ such that $V(G_{W})=W$ and $E(G_{W})=\{\{x_{i},x_{j}\}\in E(G):x_{i},x_{j}\in W\}$. For $x_{v}\in V(G)$, let $N_{G}(x_{v})$ denote the neighborhood of $x_{v}$ and let $N_{G}[x_{v}]=N_{G}(x_{v})\cup\{x_{v}\}$. Then $I(G)+(x_{v})=(x_{v})+I\left(G_{V(G)\setminus\{x_{v}\}}\right)$ and $I(G):(x_{v})=\left(x_{i}:x_{i}\in N_{G}(x_{v})\right)+I\left(G_{V(G)\setminus N_{G}[x_{v}]}\right)$. Hence

[TABLE]

[TABLE]

Thus, by virtue of Lemma 1.4, it follows that

Lemma 1.5**.**

[TABLE]

The following lemma is somewhat technical.

Lemma 1.6**.**

Let $G$ be a finite simple graph and let $x_{v}\in V(G)$. Assume that

1. (1)

$\displaystyle\deg h\left({\frac{K[V(G)\setminus\{x_{v}\}]}{I\left(G_{V(G)\setminus\{x_{v}\}}\right)},\ \lambda}\right)<\dim\frac{K[V(G)\setminus\{x_{v}\}]}{I\left(G_{V(G)\setminus\{x_{v}\}}\right)}=:d$* ;* 2. (2)

$\displaystyle\deg h\left({\frac{K[V(G)\setminus N_{G}[x_{v}]\ ]}{I\left(G_{V(G)\setminus N_{G}[x_{v}]}\right)},\ \lambda}\right)=\dim\frac{K[V(G)\setminus N_{G}[x_{v}]\ ]}{I\left(G_{V(G)\setminus N_{G}[x_{v}]}\right)}=:d^{\prime}$* ;* 3. (3)

$d>d^{\prime}$.

Then $\displaystyle\deg h\left(K[V(G)]/I(G),\ \lambda\right)=\dim K[V(G)]/I(G)=d$.

Proof.

It follows from Lemma 1.5.

By using Lemma 1.5 again, one has the Hilbert series of $K[V(G)]/I(G)$ when $G$ is a star graph or a star triangle. For $s\geq 1$, we denote by $G^{{\rm star}(x_{v})}_{s}$, the star graph joining $s$ paths of length $1$ at the common vertex $x_{v}$; see Figure 2.

Lemma 1.7**.**

Let $s\geq 1$ be an integer. Then

[TABLE]

In particular,

[TABLE]

For $t\geq 1$, we denote by $G^{{\triangle}(x_{v})}_{t}$, the star triangle joining $t$ triangles at the common vertex $x_{v}$; see Figure 3.

Lemma 1.8**.**

Let $t\geq 1$ be an integer. Then

[TABLE]

In particular,

[TABLE]

and $\displaystyle\dim K[V(G^{\triangle(x_{v})}_{t})]/I(G^{\triangle(x_{v})}_{t})=t$.

We also use the following lemmata.

Lemma 1.9** ([14, Lemma 1.5(i)]).**

Let $S_{1}$ and $S_{2}$ be polynomial rings over a field $K$. Let $I_{1}$ be a nonzero homogeneous ideal of $S_{1}$ and $I_{2}$ that of $S_{2}$. Write $S$ for $S_{1}\otimes_{K}S_{2}$ and regard $I_{1}+I_{2}$ as homogeneous ideals of $S$. Then

[TABLE]

In particular,

[TABLE]

[TABLE]

Let $G$ be a disconnected graph whose connected components are $G_{1},\ldots,G_{r}$. Then $I(G)=\sum_{i=1}^{r}I(G_{i})$. Thus, by virtue of Lemma 1.9, one has

Lemma 1.10**.**

Under the notation as above,

[TABLE]

[TABLE]

here we regard $K[V(G_{i})]/I(G_{i})$ as a 1-dimensional polynomial ring if $G_{i}$ is an isolated vertex.

Now we are in the position to prove Proposition 1.3.

Proof of Proposition 1.3.

Let $G$ be a Cameron–Walker graph as in Figure 1. We prove the equality (1.1) by using induction on $m+n$.

First, we assume that $m+n=2$. Then $m=n=1$. If $t_{1}=0$, then $G=G^{{\rm star}(v_{1})}_{s_{1}+1}$. Hence the equality (1.1) follows by Lemma 1.7. Next assume $t_{1}>0$ . We will show

[TABLE]

Note that

• •

$G_{V(G)\setminus\{v_{1}\}}$ consists of $s_{1}$ isolated vertices and a star triangle $G^{\triangle(w_{1})}_{t_{1}}$;

• •

$G_{V(G)\setminus N_{G}[v_{1}]}$ consists of $t_{1}$ star graphs $G^{{\rm star}(y^{(1)}_{1,1})}_{1},\ldots,G^{{\rm star}(y^{(1)}_{t_{1},1})}_{1}$;

see Figure 4.

Hence, by using Lemmata 1.7, 1.8 and 1.9, one has

[TABLE]

and

[TABLE]

Thus, by virtue of Lemma 1.5, it follows that

[TABLE]

Therefore one has $\deg h\left(K[V(G)]/I(G),\ \lambda\right)=\dim K[V(G)]/I(G)=s_{1}+t_{1}$, as desired.

Next, we assume that $m+n>2$.

(First Step.) Let $m=1$ and $n>1$. Suppose that there exists $1\leq\ell\leq n$ such that $t_{\ell}=0$. We may assume $\ell=n$. Then we will show

[TABLE]

Since $t_{n}=0$, $\{v_{1},w_{n}\}$ is a leaf edge. Hence we can regard $G$ as a Cameron–Walker graph such that its bipartite part is the star graph $G^{{\rm star}(v_{1})}_{n-1}$ and the vertex $v_{1}$ has $s_{1}+1$ leaf edges. Thus, by induction hypothesis, one has

[TABLE]

as desired.

Next, suppose that $t_{j}>0$ for all $1\leq j\leq n$. We will show

[TABLE]

Note that

• •

$G_{V(G)\setminus\{v_{1}\}}$ consists of $s_{1}$ isolated vertices and $n$ star triangles $G^{\triangle(w_{1})}_{t_{1}},\ldots,G^{\triangle(w_{n})}_{t_{n}}$,

• •

$G_{V(G)\setminus N_{G}[v_{1}]}$ consists of $\sum_{j=1}^{n}t_{j}$ star graphs $G^{{\rm star}(y^{(k)}_{\ell,1})}_{1}$ for $1\leq k\leq n$ and $1\leq\ell\leq t_{k}$;

see Figure 5.

Hence, by using Lemmata 1.7, 1.8 and 1.9, one has

[TABLE]

and

[TABLE]

Thus, by virtue of Lemma 1.5, it follows that

[TABLE]

Therefore one has

[TABLE]

as desired.

(Second Step.) Let $m>1$ and $n=1$. We will show

[TABLE]

Note that

• •

$G_{V(G)\setminus\{w_{1}\}}$ consists of $m+t_{1}$ star graphs $G^{{\rm star}(v_{1})}_{s_{1}},\ldots,G^{{\rm star}(v_{m})}_{s_{m}}$ and $G^{{\rm star}(y^{(1)}_{1,1})}_{1},\ldots,G^{{\rm star}(y^{(1)}_{t_{1},1})}_{1}$,

• •

$G_{V(G)\setminus N_{G}[w_{1}]}$ consists of $\sum_{i=1}^{m}s_{i}$ isolated vertices;

see Figure 6.

Hence, by using Lemmata 1.7 and 1.9, one has

[TABLE]

and

[TABLE]

Thus, by virtue of Lemma 1.5, it follows that

[TABLE]

Hence $\deg h\left(K[V(G)]/I(G),\ \lambda\right)=\sum_{i=1}^{m}s_{i}+t_{1}+\max\{t_{1},1\}-t_{1}=\sum_{i=1}^{m}s_{i}+\max\{t_{1},1\}$. Therefore, one has

[TABLE]

as desired.

(Third Step.) Let $m>1$ and $n>1$. Suppose that there exists $1\leq\ell\leq n$ such that $\{v_{m},w_{\ell}\}$ is a leaf edge. We may assume $\ell=n$. Then $t_{n}=0$. We will show

[TABLE]

Note that we can regard $G$ as a Cameron–Walker graph such that its bipartite part has bipartition $\{v_{1},\ldots,v_{m}\}\cup\{w_{1},\ldots,w_{n-1}\}$, the vertex $v_{i}$ has $s_{i}$ leaf edges for all $1\leq i\leq m-1$ and the vertex $v_{m}$ has $s_{m}+1$ leaf edges. Thus, by induction hypothesis, one has

[TABLE]

as desired.

Next, suppose that $\{v_{m},w_{\ell}\}$ is not a leaf edge for all $1\leq\ell\leq n$. Then $G_{V(G)\setminus\{v_{m}\}}$ consists of

1. (a1)

$s_{m}$ isolated vertices $x_{1}^{(m)},\ldots,x_{s_{m}}^{(m)}$; 2. (a2)

star graphs $G^{{\rm star}(v_{i})}_{s_{i}+\alpha_{i}}$ for $1\leq i\leq m-1$ with $N(v_{i})\cap\{w_{1},\ldots,w_{n}\}=:\{w_{j_{1}},\ldots,w_{j_{\alpha_{i}}}\}$ satisfying $N(w_{j_{k}})\subset\{v_{i},v_{m}\}$ for any $k=1,\ldots,\alpha_{i}$; 3. (a3)

star triangles $G^{\triangle(w_{j})}_{t_{j}}$ for $1\leq j\leq n$ with $N(w_{j})\cap\{v_{1},\ldots,v_{m}\}=\{v_{m}\}$; 4. (a4)

some Cameron–Walker induced subgraphs.

We give an example after the proof; see Example 1.11.

Note that each graph of type (a2) can be considered as a Cameron–Walker induced subgraph. Also note that each induced star graph $G^{{\rm star}(v_{i})}_{s_{i}}$ (resp.  induced pendant triangle $G^{\triangle(w_{j})}_{t_{j}}$) appears in (a2) or (a4) (resp.  (a3) or (a4)) as a (sub)graph. Hence by virtue of Lemmata 1.7, 1.8, 1.10 and induction hypothesis, one has

[TABLE]

and

[TABLE]

On the other hand, $G_{V(G)\setminus N_{G}[v_{m}]}$ consists of

1. (b1)

star graphs $G^{{\rm star}(v_{i})}_{s_{i}}$ for $1\leq i\leq m-1$ with $N(v_{i})\cap\{w_{1},\ldots,w_{n}\}\subset N(v_{m})$; 2. (b2)

star graphs $G^{{\rm star}(y^{(j)}_{\ell,1})}_{1}$ for $1\leq j\leq n$ with $\{v_{m},w_{j}\}\in E(G)$ and $1\leq\ell\leq t_{j}$. 3. (b3)

some Cameron–Walker induced subgraphs;

see Example 1.11.

Note that each induced star graph $G^{{\rm star}(v_{i})}_{s_{i}}$ appears in (b1) or (b3) as a (sub)graph. Also note that the star graphs $G^{{\rm star}(y^{(j)}_{\ell,1})}_{1}$, $1\leq\ell\leq t_{j}$ of type (b2) are the edges of the pendant triangle $G^{\triangle(w_{j})}_{t_{j}}$ and the total contributions of these graphs to the degree of $h$-polynomial and the dimension are both $t_{j}$. Hence, by virtue of Lemmata 1.7, 1.10 and induction hypothesis, it follows that

[TABLE]

Thus Lemma 1.6 says that

[TABLE]

as desired.

We give an example of Cameron–Walker graph with $m>1$ and $n>1$ which would be helpful to understand (Third Step.) of the proof of Proposition 1.3.

Example 1.11**.**

Let $G$ be the following Cameron–Walker graph:

$G=$$\textstyle{v_{m}=v_{3}}$

Then the induced subgraph $G_{V(G)\setminus\{v_{m}\}}$ is as follows.

$G_{V(G)\setminus\{v_{m}\}}=$

Also the induced subgraph $G_{V(G)\setminus N_{G}[v_{m}]}$ is as follows.

$G_{V(G)\setminus N_{G}[v_{m}]}=$

2. Cameron–Walker graphs with the equality $(*)$

As noted in Introduction, for an arbitrary finite simple graph $G$, one has

[TABLE]

where we set $S=K[V(G)]$. Then it is natural to ask for which graph $G$ satisfies the equality:

[TABLE]

Recall that the equality $(\ast)$ holds if and only if $S/I(G)$ has a unique extremal Betti number. Hence when $I(G)$ has a pure resolution ([5, p.  153]), the equality $(\ast)$ holds. Moreover by ([2, Lemma 3]), it follows that the equality $(\ast)$ holds if $S/I(G)$ is Cohen–Macaulay.

In this section, we give a classification of Cameron–Walker graphs $G$ with the equality $(*)$.

Throughout this section, let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1. By Theorem 1.1, the equality $(*)$ holds if and only if $\displaystyle\operatorname{depth}\left(S/I(G)\right)=\operatorname{reg}\left(S/I(G)\right)$. Both of these invariants have combinatorial explanations. The regularity is equal to the induced matching number (or the matching number) of $G$: $\operatorname{reg}\left(S/I(G)\right)=\sum_{j=1}^{n}t_{j}+m$. In order to state about the depth, we need some definitions.

For a subset $A\subset V(G)$, we set $N_{G}(A)=\bigcup_{v\in A}N_{G}(v)\setminus A$. A subset $A\subset V(G)$ is said to be independent if $\{x_{i},x_{j}\}\notin E(G)$ for any $x_{i},x_{j}\in A$. We denote by $i(G)$, the minimum cardinality of independent sets $A$ with $A\cup N_{G}(A)=V(G)$. Then $\displaystyle\operatorname{depth}\left(S/I(G)\right)=i(G)$; see [11, Corollary 3.7].

We have the following estimation for $i(G)$.

Lemma 2.1**.**

Let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1. Then

[TABLE]

Moreover if the bipartite part of $G$ is the complete bipartite graph, then

[TABLE]

Proof.

The upper bound is clear. We prove the lower bound.

Let $A\subset V(G)$ be an independent set with $A\cup N_{G}(A)=V(G)$. Then we put $A_{\rm bip}=A\cap\{v_{1},\ldots,v_{m},w_{1},\ldots,w_{n}\}$ and $A^{\prime}=A\setminus A_{\rm bip}$. We note that $A=A_{\rm bip}\sqcup A^{\prime}$, and

• •

If $v_{i}\not\in A_{\rm bip}$, then $x^{(i)}_{1},\ldots,x^{(i)}_{s_{i}}\in A^{\prime}$;

• •

If $w_{j}\not\in A_{\rm bip}$, then $y^{(j)}_{\ell,1}\in A^{\prime}$ or $y^{(j)}_{\ell,2}\in A^{\prime}$ for all $1\leq\ell\leq t_{j}$.

Hence one has

[TABLE]

Thus $i(G)\geq m+|\{j:t_{j}>0\}|$.

When the bipartite part of $G$ is the complete bipartite graph, one has either $A_{\rm bip}\subset\{v_{1},\ldots,v_{m}\}$ or $A_{\rm bip}\subset\{w_{1},\ldots,w_{n}\}$. For the former case, since $s_{i}\geq 1$ for all $i$, it follows that $|A|\geq\sum_{j=1}^{n}t_{j}+m$. For the latter case, one has $|A|\geq\sum_{i=1}^{m}s_{i}+n$ because $w_{j}\in A_{\rm bip}$ if $t_{j}=0$. It then follows that

[TABLE]

Combining this with the upper bound, one has the equality.

By virtue of this lemma, we can give a classification of Cameron–Walker graphs $G$ satisfying the equality $(*)$.

Theorem 2.2**.**

Let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1 and $G_{\rm bip}$ the bipartite part of $G$. Then $S/I(G)$ satisfies the equality $(*)$ if and only if

[TABLE]

holds for all $V\subset\{v_{1},\ldots,v_{m}\}$.

Proof.

Assume that there exists a subset $V\subset\{v_{1},\ldots,v_{m}\}$ satisfying

[TABLE]

Let

[TABLE]

[TABLE]

Then $A$ is an independent set with $A\cup N_{G}(A)=V(G)$ and

[TABLE]

Hence we have

[TABLE]

Thus $S/I(G)$ does not satisfy the equality $(*)$.

Next, we assume that

[TABLE]

holds for all $V\subset\{v_{1},\ldots,v_{m}\}$.

Let $A$ be an independent set of $V(G)$ with $A\cup N_{G}(A)=V(G)$. Let $A_{v}=A\cap\{v_{1},\ldots,v_{m}\}$ and $A_{w}=A\cap\{w_{1},\ldots,w_{n}\}$. Then,

[TABLE]

For $j$ satisfying $N_{G_{\rm bip}}(w_{j})\subset\{v_{1},\ldots,v_{m}\}\setminus A_{v}$ and $w_{j}\not\in A_{w}$, one has $t_{j}\geq 1$. Hence

[TABLE]

Since $N_{G_{\rm bip}}(w_{j})\subset\{v_{1},\ldots,v_{m}\}\setminus A_{v}$ if $w_{j}\in A_{w}$, one has

[TABLE]

Considering the inequality (2.1) for $V=\{v_{1},\ldots,v_{m}\}\setminus A_{v}$, it follows that

[TABLE]

Hence we have

[TABLE]

Thus one has

[TABLE]

This inequality together with Lemma 2.1 says that

[TABLE]

Therefore $S/I(G)$ satisfies the equality $(*)$.

Remark 2.3**.**

1. (1)

When we use Theorem 2.2, we only need to check the inequality (2.1) for $V\subset\{v_{1},\ldots,v_{m}\}$ with $N_{G_{\rm bip}}(w_{j})\subset V$ for some $1\leq j\leq n$. Indeed, let $V$ be a subset of $\{v_{1},\ldots,v_{m}\}$ such that $N_{G_{\rm bip}}(w_{j})\not\subset V$ for all $1\leq j\leq n$. Then the inequality (2.1) for $V$ is $\sum_{1\leq i\leq m,v_{i}\in V}s_{i}\geq|V|$, which always holds since $s_{i}\geq 1$ for all $1\leq i\leq m$. 2. (2)

Considering the inequality (2.1) for $V=\{v_{1},\ldots,v_{m}\}$, it follows that $\sum_{i=1}^{m}s_{i}+n\geq\sum_{j=1}^{n}t_{j}+m$ holds if $S/I(G)$ satisfies the equality $(*)$.

As a corollary of Theorem 2.2, one has

Corollary 2.4**.**

Let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1. Suppose that $t_{j}\leq 1$ for all $1\leq j\leq n$. Then $S/I(G)$ satisfies the equality $(*)$.

Remark 2.5**.**

Let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1. Then $S/I(G)$ is Cohen-Macaulay if and only if $s_{i}=1$ for all $1\leq i\leq m$ and $t_{j}=1$ for all $1\leq j\leq n$ ([11, Theorem 1.3]). Hence the class of graphs in Corollary 2.4 contains all Cohen–Macaulay Cameron–Walker graphs.

Proof of Corollary 2.4.

Since $s_{i}\geq 1$ for all $1\leq i\leq m$ and $t_{j}\leq 1$ for all $1\leq j\leq n$, one has

[TABLE]

for all $V\subset\{v_{1},\ldots,v_{m}\}$. Hence $S/I(G)$ satisfies the equality $(*)$ by Theorem 2.2.

From Theorem 2.2, we also have

Corollary 2.6**.**

Let $G$ be a Cameron–Walker graph whose bipartite part is the complete bipartite graph. We label the vertices of $G$ as in Figure 1. Then $S/I(G)$ satisfies the equality $(*)$ if and only if $\sum_{i=1}^{m}s_{i}+n\geq\sum_{j=1}^{n}t_{j}+m$.

Proof.

Since $N_{G_{\rm bip}}(w_{j})=\{v_{1},\ldots,v_{m}\}$ for all $1\leq j\leq n$, the claim follows from Theorem 2.2 and Remark 2.3.

In general, one has $\dim S/I(G)\geq\operatorname{depth}(S/I(G))$. Then it is natural to ask the following

Question 2.7**.**

Given arbitrary integers $d,e$ with $d\geq e\geq 1$, are there a Cameron–Walker graph $G$ satisfying $\dim S/I(G)=d$ and $\operatorname{depth}(S/I(G))=e$?

As an application of Corollary 2.4, we give a complete answer for Question 2.7.

We first note about the depth.

Proposition 2.8**.**

Let $G$ be a Cameron–Walker graph. Then $\operatorname{depth}S/I(G)\geq 2$. Moreover $\operatorname{depth}S/I(G)=2$ if and only if $G$ can be considered as one of the following Cameron–Walker graphs:

1. (e1)

$m=2$* and $t_{j}=0$ for all $1\leq j\leq n;$* 2. (e2)

$m=n=1$* and $t_{1}=1;$* 3. (e3)

$m=n=1$, $t_{1}\geq 2$, and $s_{1}=1.$

Here, we use labeling of vertices of $G$ as in Figure 1.

Proof.

Assume that $G$ is a Cameron–Walker graph with $\operatorname{depth}(S/I(G))=1$. By Lemma 2.1, one has $m=1$ and $t_{j}=0$ for all $1\leq j\leq n$. Then $G$ is a star graph but this is a contradiction since star graphs are not Cameron–Walker by definition.

Next assume that $G$ is a Cameron–Walker graph with $\operatorname{depth}(S/I(G))=2$. By Lemma 2.1, one has

• •

$m=2$ and $t_{j}=0$ for all $1\leq j\leq n$, or

• •

$m=1$ and $t_{j}=0$ except for one $j$.

We consider the case $m=1$. Since $G$ is not a star graph, there exists just one $j$ with $t_{j}\neq 0$, say $j=1$. When $n\geq 2$, since $m=1$ and $t_{j}=0$ for $2\leq j\leq n$, $G$ can be considered as a Cameron–Walker graph whose bipartite subgraph is of type $(1,1)$ such that $v_{1}$ has $s_{1}+(n-1)$ leaf edges and $w_{1}$ has one pendant triangle. Thus we may assume $n=1$. If $t_{1}\geq 2$, then $i(G)=\operatorname{depth}S/I(G)=2$ implies that $s_{1}=1$. Hence the assertion follows.

The converse is easy.

Since any Cameron–Walker graph $G$ satisfies $\operatorname{depth}S/I(G)\geq 2$, we only consider the case $e\geq 2$ in Question 2.7. By virtue of Corollary 2.4, we can give a Cameron–Walker graph $G$ satisfying the properties in Question 2.7 with the equality $(\ast)$.

Corollary 2.9**.**

Given arbitrary integers $d,e$ with $d\geq e\geq 2$, there exists a Cameron–Walker graph $G$ with the equality $(\ast)$ satisfying $\dim S/I(G)=d$ and $\operatorname{depth}(S/I(G))=e$.

Proof.

We use the labeling of vertices of a Cameron–Walker graph as in Figure 1.

$\bullet$** The case $d>e$:** Let $G$ be the Cameron–Walker graph with $m=e$, $n=1$, $s_{1}=\cdots=s_{e-1}=1$, $s_{e}=d-e$, and $t_{1}=0$. Then $\dim(S/I(G))=\sum_{i=1}^{e}s_{i}+\max\{t_{1},1\}=d$. Also, $A:=\{v_{1},\ldots,v_{e}\}$ is an independent set of $V(G)$ with $A\cup N_{G}(A)=V(G)$ which gives $i(G)$. Thus one has $\operatorname{depth}S/I(G)=i(G)=|A|=e$.

$\bullet$** The case $d=e$:** Let $G$ be the Cameron–Walker graph with $m=d-1$, $n=1$, $s_{1}=\cdots=s_{d-1}=1$, and $t_{1}=1$. Then $\dim(S/I(G))=\sum_{i=1}^{d-1}s_{i}+\max\{t_{1},1\}=d$. Also, $A:=\{x_{1}^{(1)},\ldots,x_{d-1}^{(1)}\}\cup\{w_{1}\}$ is an independent set of $V(G)$ with $A\cup N_{G}(A)=V(G)$ which gives $i(G)$. Thus one has $\operatorname{depth}S/I(G)=i(G)=|A|=e$.

Finally of the section, we provide some classes of graphs $G$ which satisfy the equality $(\ast)$ other than Cameron–Walker graphs.

Proposition 2.10**.**

Let $G$ be the one of the following graph. Then the equality $(\ast)$ satisfies:

1. (1)

The star graph $G^{{\rm star}(x_{v})}_{s}$ $(s\geq 1)$. 2. (2)

The path graph $P_{n}$ $(n\geq 2)$. 3. (3)

The $n$-cycle $C_{n}$ $(n\geq 3)$. 4. (4)

The graph $G_{s}$ on $\{x_{1},\ldots,x_{s+4}\}$ where $s\geq 1$ which consists of the star graph $G^{{\rm star}(x_{s+3})}_{s}$ on $\{x_{1},\ldots,x_{s}\}\cup\{x_{s+3}\}$ and $P_{4}$ on $\{x_{s+1},\ldots,bx_{s+4}\}$; see Figure 7.

Before proving Proposition 2.10, we recall some facts on invariants of an edge ideal. For a finite simple graph $G$, the dimension $\dim S/I(G)$ is equal to the maximum cardinality of independent sets of $G$. In particular, one has $\dim S/I(P_{n})=\lceil n/2\rceil$ and $\dim S/I(C_{n})=\lceil(n-1)/2\rceil$.

We also recall the non-vanishing theorem of Betti numbers of edge ideals.

Lemma 2.11** ([16, Theorems 3.1 and 4.1]).**

Let $G$ be a finite simple graph. Suppose that there exists a set of star subgraphs $\{B_{1},\ldots,B_{\ell}\}\ (\ell\geq 1)$ of $G$ satisfying the following conditions$:$

1. (1)

$V(B_{k})\cap V(B_{{k}^{\prime}})=\emptyset$* for all $1\leq k<{k}^{\prime}\leq\ell;$* 2. (2)

There exist edges $e_{1},\ldots,e_{\ell}$ with $e_{k}\in E(B_{k})$, $k=1,\ldots,\ell$ such that $\{e_{1},\ldots,e_{\ell}\}$ forms an induced matching of $G$.

Set $B_{k}=G^{{\rm star}(x_{\beta_{k}})}_{\alpha_{k}}$ $(1\leq k\leq\ell)$ and $i=\alpha_{1}+\cdots+\alpha_{\ell}$. Then one has

[TABLE]

Moreover, when $G$ has no cycle, $\beta_{i,i+\ell}(S/I(G))\neq 0$ if and only if there exists such a set of star subgraphs of $G$.

By Lemma 2.11, it follows that the equality $\operatorname{reg}(S/I(G))=im(G)$ holds when $G$ has no cycle, which was first proved by Zheng [22].

Now we prove Proposition 2.10.

Proof of Proposition 2.10.

Recall that the equality $(\ast)$ is satisfied if and only if $(p,p+r)$-th Betti number does not vanish where $p$ is the projective dimension and $r$ is the regularity.

1. (1)

Since $G^{{\rm star}(x_{v})}_{s}$ has no cycle, one has $\operatorname{reg}(S/I(G^{{\rm star}(x_{v})}_{s}))=im(G)=1$ by [22]. Also, it is easy to see from Lemma 2.11 that $\operatorname{proj\,dim}(S/I(G^{{\rm star}(x_{v})}_{s}))=s$, and $\beta_{s,s+1}(S/I(G^{{\rm star}(x_{v})}_{s}))\neq 0$. 2. (2)

Let $V(P_{n})=\{x_{1},x_{2},\ldots x_{n}\}$ and $E(P_{n})=\left\{\{x_{1},x_{2}\},\{x_{2},x_{3}\},\ldots,\{x_{n-1},x_{n}\}\right\}$. It follows from [18, Lemma 2.8] that $\operatorname{depth}(S/I(P_{n}))=\lceil n/3\rceil$. Hence by Auslander–Buchsbaum Theorem, one has

[TABLE]

Also, by [4, p.4, Proposition], one has

[TABLE]

• •

**The case $n=3\ell$ or $n=3\ell+1$ : ** Then $p=2\ell$ and $r=\ell$. For $1\leq k\leq\ell$, let $B_{k}$ be the induced subgraph of $P_{n}$ on $\{x_{3(k-1)+1},x_{3(k-1)+2},x_{3k}\}$. Then $B_{k}$ is the star subgraph $G^{{\rm star}(x_{3(k-1)+2})}_{2}$. Take $e_{k}:=\{x_{3(k-1)+1},x_{3(k-1)+2}\}\in E(B_{k})$. Then $\{e_{1},\ldots,e_{\ell}\}$ forms an induced matching of $P_{n}$. Thus Lemma 2.11 says that $\beta_{p,p+r}(S/I(P_{n}))=\beta_{2\ell,2\ell+\ell}(S/I(P_{n}))\neq 0$.

• •

**The case $n=3\ell+2$ : ** Then $p=2\ell+1$ and $r=\ell+1$. For $1\leq k\leq\ell$, let $B_{k}$ be the induced subgraph of $P_{n}$ on $\{x_{3(k-1)+1},x_{3(k-1)+2},x_{3k}\}$. Then $B_{k}$ is the star subgraph $G^{{\rm star}(x_{3(k-1)+2})}_{2}$. Also let $B_{\ell+1}$ be the induced subgraph of $P_{n}$ on $\{x_{3\ell+1},x_{3\ell+2}\}$, which is the star subgraph $G^{{\rm star}(x_{3\ell+2})}_{1}$. Take $e_{k}:=\{x_{3(k-1)+1},x_{3(k-1)+2}\}\in E(B_{k})$ for $k=1,\ldots,\ell,\ell+1$. Then $\{e_{1},\ldots,e_{\ell},e_{\ell+1}\}$ forms an induced matching of $P_{n}$. Thus Lemma 2.11 says that $\beta_{p,p+r}(S/I(P_{n}))=\beta_{2\ell+1,(2\ell+1)+\ell+1}(S/I(P_{n}))\neq 0$. 3. (3)

Let $V(C_{n})=\{x_{1},x_{2},\ldots x_{n}\}$ and $E(C_{n})=\left\{\{x_{1},x_{2}\},\ldots,\{x_{n-1},x_{n}\},\{x_{1},x_{n}\}\right\}$. It follows from [7, p.  117] that

[TABLE]

Hence by Auslander–Buchsbaum Theorem, one has

[TABLE]

Also by [1, Theorem 5.2], one has

[TABLE]

Then we can prove the case where $n=3\ell$. In this case, $p=2\ell$ and $r=\ell$. For $1\leq k\leq\ell$, let $B_{k}$ be the induced subgraph of $C_{n}$ on $\{x_{3(k-1)+1},x_{3(k-1)+2},x_{3k}\}$. Then $B_{k}$ is the star subgraph $G^{{\rm star}(x_{3(k-1)+2})}_{2}$. Take $e_{k}:=\{x_{3(k-1)+1},x_{3(k-1)+2}\}\in E(B_{k})$. Then $\{e_{1},\ldots,e_{\ell}\}$ forms an induced matching of $C_{n}$. Thus Lemma 2.11 says that $\beta_{p,p+r}(S/I(C_{n}))=\beta_{2\ell,2\ell+\ell}(S/I(C_{n}))\neq 0$. Hence $S/I(C_{n})$ satisfies the equality $(*)$.

For the cases $n=3\ell+1,3\ell+2$, we compute all invariants appearing in the equality $(\ast)$. We have already known the dimension, the depth, and the regularity. In order to compute $\deg h(S/I(C_{n}),\lambda)$, consider the short exact sequence

[TABLE]

Since $I(C_{n})+(x_{n})=(x_{n})+I(P_{n-1})$, we have

[TABLE]

Also since $I(C_{n}):(x_{n})=(x_{1},x_{n-1})+(x_{2}x_{3},\ldots,x_{n-3}x_{n-2})$, we have

[TABLE]

Thus Lemma 1.4 says that

[TABLE]

By (2), one has

[TABLE]

• •

**The case $n=3\ell+1$ : ** Then $\operatorname{reg}(S/I(C_{n}))=\operatorname{depth}(S/I(C_{n}))=\ell$ and $\dim S/I(C_{n})=\lceil 3\ell/2\rceil$. Moreover, since

[TABLE]

and

[TABLE]

one has $\deg h(S/I(C_{n}),\lambda)=\lceil 3\ell/2\rceil$. Hence $S/I(C_{n})$ satisfies the equality $(*)$.

• •

**The case $n=3\ell+2$ : ** Then $\operatorname{reg}(S/I(C_{n}))=\operatorname{depth}(S/I(C_{n}))=\ell+1$ and $\dim S/I(C_{n})=\lceil(3\ell+1)/2\rceil$. Moreover, since

[TABLE]

and

[TABLE]

one has $\deg h(S/I(C_{n}),\lambda)=\lceil(3\ell+1)/2\rceil$. Hence $S/I(C_{n})$ satisfies the equality $(*)$. 4. (4)

Since $G_{s}$ has no cycle, one has $\operatorname{reg}(S/I(G_{s}))=im(G)=1$ by [22]. Also it is easy to see from Lemma 2.11 that $\operatorname{proj\,dim}(S/I(G_{s}))=s+2$, and $\beta_{s+2,(s+2)+1}(S/I(G_{s}))\neq 0$.

Remark 2.12**.**

The graph $G_{s}$ in Proposition 2.10 (as well as $P_{3\ell+1}$) is an example of a graph satisfying $(\ast)$ with $\deg h(S/I(G_{s}),\lambda)<\dim S/I(G_{s})(=s+2)$ because $\operatorname{reg}(S/I(G_{s}))=1<2=(s+4)-\operatorname{proj\,dim}(S/I(G_{s}))=\operatorname{depth}(S/I(G_{s}))$. Note that Cameron–Walker graphs $G$ satisfies $\deg h(S/I(G),\lambda)=\dim S/I(G)$.

3. Other properties on Cameron–Walker graphs

In this section, we provide some properties on a Cameron–Walker graph derived from the results of previous sections.

Let $G$ be a finite simple graph and $S=K[V(G)]$. Suppose that $S/I(G)$ is Cohen–Macaulay. Then the equalities $(*)$ and $\dim S/I(G)=\operatorname{depth}(S/I(G))$ hold. Hence one has $\deg h(S/I(G),\lambda)=\operatorname{reg}(S/I(G))$. Nevertheless, $\deg h(S/I(G),\lambda)=\operatorname{reg}(S/I(G))$ does not imply that $S/I(G)$ is Cohen–Macaulay, see [12, Example 3.2]. Moreover, in general, there is no relationship between the regularity and the degree of the $h$-polynomial. Actually, [13] proved that for given integers $r,s\geq 1$, there exists a finite simple graph $G$ such that $\operatorname{reg}(S/I(G))=r$ and $\deg h(S/I(G),\lambda)=s$. However, we can derive from Proposition 1.3 the relation between $\operatorname{reg}(S/I(G))$ and $\deg h(S/I(G),\lambda)$ when $G$ is Cameron–Walker. Moreover we provide a complete classification of Cameron–Walker graphs $G$ with $\deg h(S/I(G),\lambda)=\operatorname{reg}(S/I(G))$.

Theorem 3.1**.**

Let $G$ be a Cameron–Walker graph whose labeling of vertices is as in Figure 1. Then we have $\deg h(S/I(G),\lambda)\geq\operatorname{reg}(S/I(G))$. Moreover the equality $\deg h(S/I(G),\lambda)=\operatorname{reg}(S/I(G))$ holds if and only if $s_{i}=1$ for all $1\leq i\leq m$ and $t_{j}\geq 1$ for all $1\leq j\leq n$.

Proof.

We first note that $\operatorname{reg}(S/I(G))=\sum_{j=1}^{n}t_{j}+m$. Combining this with Proposition 1.3, one has

[TABLE]

Note that each summands of right hand-side is non-negative. Then the desired assertion follows.

Let $G$ be a Cameron–Walker graph. Combining the inequality

[TABLE]

with Theorem 1.1, Theorem 3.1, and Proposition 2.8, one has

[TABLE]

Then it is natural to ask the following

Question 3.2**.**

Given arbitrary integers $d,r,e$ with $d\geq r\geq e\geq 2$, is there a Cameron–Walker graph $G$ satisfying

[TABLE]

We have already investigated Cameron–Walker graphs $G$ with $\operatorname{depth}S/I(G)=2$ in Proposition 2.8. Their invariants are as follows:

1. (e1)

$\dim S/I(G)=\deg h\left(S/I(G),\lambda\right)=s_{1}+s_{2}+n>2=\operatorname{reg}(S/I(G))=\operatorname{depth}(S/I(G))$. 2. (e2)

$\dim S/I(G)=\deg h\left(S/I(G),\lambda\right)=s_{1}+1\geq 2=\operatorname{reg}(S/I(G))=\operatorname{depth}(S/I(G))$. 3. (e3)

$\dim S/I(G)=\deg h\left(S/I(G),\lambda\right)=\operatorname{reg}(S/I(G))=t_{1}+1>2=\operatorname{depth}(S/I(G))$.

Therefore we have the following answer for Question 3.2 when $e=2$.

Corollary 3.3**.**

Let $d,r,e$ be integers with $d\geq r\geq e=2$. Then there exists a Cameron–Walker graph $G$ satisfying $(\ast\ast)$ if and only if $r=2$ or $r=d$.

When $e\geq 3$, we have the following answer for Question 3.2.

Theorem 3.4**.**

Given arbitrary integers $d,r,e$ with $d\geq r\geq e\geq 3$, there exists a Cameron–Walker graph $G$ satisfying $\dim S/I(G)=\deg h(S/I(G),\lambda)=d$, $\operatorname{reg}(S/I(G))=r$, and $\operatorname{depth}(S/I(G))=e$.

Proof.

We use the labeling of vertices of a Cameron–Walker graph as in Figure 1. Set $V_{\rm bip}=\{v_{1},\ldots,v_{m},\;w_{1},\ldots,w_{n}\}$.

$\bullet$** The case $d>r$:** Let $G$ be the Cameron–Walker graph with $m=e-1$, $n=2$, $s_{1}=\cdots=s_{e-2}=1$, $s_{e-1}=d-r$, $t_{1}=r-e+1$, and $t_{2}=0$ such that

[TABLE]

see Figure 8.

Then it is easy to see that $\dim(S/I(G))=\deg h\left(S/I(G),\lambda\right)=d$ and $\operatorname{reg}(S/I(G))=r$. Also, $A:=\{v_{2},\ldots,v_{e-1}\}\cup\{x_{1}^{(1)},w_{1}\}$ is an independent set of $V(G)$ with $A\cup N_{G}(A)=V(G)$ which gives $i(G)$. Thus one has $\operatorname{depth}S/I(G)=i(G)=|A|=e$.

$\bullet$** The case $d=r$:** Let $G$ be the Cameron–Walker graph with $m=e-1$, $n=1$, $s_{1}=\cdots=s_{e-1}=1$, and $t_{1}=d-e+1$. Then it is easy to see that $\dim(S/I(G))=\deg h\left(S/I(G),\lambda\right)=\operatorname{reg}(S/I(G))=d$. Also $A:=\{x_{1}^{(1)},\ldots,x_{e-1}^{(1)}\}\cup\{w_{1}\}$ is an independent set of $V(G)$ with $A\cup N_{G}(A)=V(G)$ which gives $i(G)$. Thus one has $\operatorname{depth}S/I(G)=i(G)=|A|=e$.

Acknowledgment. The authors were partially supported by JSPS KAKENHI 26220701, 15K17507, 17K14165 and 16J01549.