Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests

TL;DR

This paper derives exact formulas for the projective dimension and regularity of powers of edge ideals in vertex-weighted rooted forests, linking algebraic invariants to vertex weights and forest structure.

Contribution

It provides novel explicit formulas for these invariants in weighted rooted forests, highlighting the influence of weights and structure on algebraic properties.

Findings

Formulas depend on vertex weights and number of edges

Examples illustrate the impact of direction choice and assumptions

Results show specific conditions where formulas hold

Abstract

In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about "rooted" forest such that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.

Equations

$$I(D)=(x_i{x}_j^{w_j}\mid x_i{x}_j\in E).$$

$$\text{mbox}reg(I(D{)}^t)=\sum _{x\in V(D)}w(x)-|E(D)|+1+(t-1)(w+1)\text{mbox}forallt\ge 1,$$

$$\text{mbox}pd(I(D{)}^t)=|E(D)|-1\text{mbox}forallt\ge 1.$$

$$I(D)=(x_i{x}_j^{w_j}\mid x_i{x}_j\in E).$$

$$\text{mbox}pd(I):=\text{mbox}max\{i|{\beta }_{i,j}(I)\ne 0\}.$$

$$\text{mbox}reg(I):=\text{mbox}max\{j-i|{\beta }_{i,j}(I)\ne 0\}.$$

$${\beta }_{i,j}(I)={\beta }_{i,j}(J)+{\beta }_{i,j}(K)+{\beta }_{i-1,j}(J\cap K)\text{mbox}foralli,j\ge 0,$$

$$\text{mbox}reg(I)=\text{mbox}max\{\text{mbox}reg(J),\text{mbox}reg(K),\text{mbox}reg(J\cap K)-1\}.$$

$$\mathcal{P}(u)=x_{11}{x}_{12}\cdots x_{1a_1}{x}_{21}\cdots x_{2a_2}\cdots x_{n1}\cdots x_{n{a}_n}$$

$$I^{\mathcal{P}}=(\mathcal{P}(u_1),\dots ,\mathcal{P}(u_m)),$$

$$\text{mbox}reg(I^t)=\sum _{i=1}^r{d}_i-(r-1)+(t-1)w\text{mbox}forallt\ge 1,$$

$$I^t=(J+(u_r))I^{t-1}=J^t+u_r{I}^{t-1}.$$

$$\varphi :I^{t-1}(-w)\oplus J^t\stackrel{\cdot (u_r,1)}{⟶}{I}^t.$$

$$0\to J^t(-w)\to I^{t-1}(-w)\oplus J^t\stackrel{\cdot (u_r,1)}{⟶}{I}^t\to 0.$$

$$\text{mbox}reg(I^{t-1}(-w))$$

$$\text{mbox}reg(J^t)=\sum _{i=1}^{r-1}{d}_i-(r-2)+(t-1)w^{\prime }$$

$$\text{mbox}reg(I^{t-1}(-w)\oplus J^t)=\text{mbox}max\{\text{mbox}reg(I^{t-1}(-w)),\text{mbox}reg(J^t)\}=\sum _{i=1}^r{d}_i-(r-1)+(t-1)w.$$

$$\text{mbox}reg(u_r{I}^{t-1}\cap J^t)=\text{mbox}reg(u_r{J}^t)=\sum _{i=1}^r{d}_i-(r-2)+(t-1)w^{\prime }.$$

$$\beta -\alpha =(t-1)(w^{\prime }-w)+1.$$

$$\text{mbox}reg(I^t)=\beta -1=\alpha =\sum _{i=1}^r{d}_i-(r-1)+(t-1)w.$$

$$\text{mbox}reg(I^t)=\alpha =\sum _{i=1}^r{d}_i-(r-1)+(t-1)w.$$

$$0\to u_{r-1}I\cap K^2\to I(-w^{\prime })\oplus K^2\stackrel{\cdot (u_{r-1},1)}{⟶}{I}^2\to 0.$$

$$\text{mbox}reg(I(-w^{\prime }))=w^{\prime }+\sum _{i=1}^r{d}_i-(r-1)=\sum _{i=1}^r{d}_i-(r-1)+w-1,$$

$$\text{mbox}reg(K^2)=\sum _{i=1}^{r-2}{d}_i+d_r-(r-2)+w=\sum _{i=1}^r{d}_i-(r-1)+2,$$

$$\text{mbox}reg(u_{r-1}{K}^2)=w^{\prime }+\sum _{i=1}^r{d}_i-(r-1)+2=\sum _{i=1}^r{d}_i-(r-1)+w+1.$$

$$(I(D{)}^t,z^{w_z})=(I(D\setminus z{)}^t,z^{w_z})\text{for any}t\ge 1.$$

$$(I(D{)}^t:y{z}^{w_z})=I(D{)}^{t-1}\text{for any}t\ge 2.$$

$$((I(D{)}^t:z^{w_z}),y)=((I(D\setminus y{)}^t:z^{w_z}),y)=(I(D\setminus y{)}^t,y)\text{for any}t\ge 2.$$

$$\text{mbox}reg(I_n)\le \sum _{i=1}^n{w}_i-(n-1)+1+(w+1)$$

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1. Introduction

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring in $n$ variables over a field $k$ and let $I\subset S$ be a homogeneous ideal. There are two central invariants associated to $I$, the regularity $\mbox{reg}\,(I):=\mbox{max}\{j-i\ |\ \beta_{i,j}(I)\neq 0\}$ and the projective dimension $\mbox{pd}\,(I):=\mbox{max}\{i\ |\ \beta_{i,j}(I)\neq 0\ \text{for some}\ j\}$, that in a sense, they measure the complexity of computing the graded Betti numbers $\beta_{i,j}(I)$ of $I$. In particular, if $I$ is a monomial ideal, its polarization $I^{\mathcal{P}}$ has the same projective dimension and regularity as $I$ and is squarefree. Thus one can associate $I^{\mathcal{P}}$ to a graph or a hypergraph or a simplicial complex. Many authors have studied the regularity and Betti numbers of edge ideals of graphs, e.g. [3, 4, 11, 23, 30, 32, 36, 39, 45, 51, 52, 53, 54]. Other authors have studied higher degree generalizations using hypergraphs and clutters [17, 18, 19, 20, 41] or simplicial complexes [22, 24].

Given any homogeneous ideal $I$, it is well known that the regularity of $I^{t}$ is asymptotically a linear function in $t$, that is, there exist constants $a$ and $b$ such that for all $t\gg 0$, $\mbox{reg}\,(I^{t})=at+b$ (see [14, 16, 40, 49]). Generally, the problem of finding the exact linear form $at+b$ and the smallest value $t_{0}$ such that $\mbox{reg}\,(I^{t})=at+b$ for all $t\geq t_{0}$ has proved to be very difficult. In [8], Brodmann showed that $\mbox{depth}\,(S/I^{t})$ is a constant for $t\gg 0$, and this constant is bounded above by $n-\ell(I)$, where $\ell(I)$ is the analytic spread of $I$. It is shown in [34, Theorem 1.2] that $\mbox{depth}\,(S/I^{t})$ is a nonincreasing function of $t$ when all powers of $I$ have a linear resolution and conditions are given in that paper under which all powers of $I$ will have linear quotients. By Auslander-Buchsbaum formula, we obtain the projective dimension $\mbox{pd}\,(S/I^{t})$ is a constant for $t\gg 0$. In this regard, there has been an interest in determining the smallest value $t_{0}$ such that $\mbox{pd}\,(S/I^{t})$ is a constant for all $t\geq t_{0}$. (see [21, 22, 26, 34, 38, 44]).

To the best of our knowledge, a few papers consider how to compute $\mbox{pd}\,(S/I^{t})$ for a homogeneous ideal $I$.

A directed graph or digraph $D$ consists of a finite set $V(D)$ of vertices, together with a collection $E(D)$ of ordered pairs of distinct points called edges or arrows. A vertex-weighted directed graph is a triplet $D=(V(D),E(D),w)$, where $w$ is a weight function $w:V(D)\rightarrow\mathbb{N}^{+}$, where $N^{+}=\{1,2,\ldots\}$. Some times for short we denote the vertex set $V(D)$ and edge set $E(D)$ by $V$ and $E$ respectively. The weight of $x_{i}\in V$ is $w(x_{i})$, denoted by $w_{i}$ or $w_{x_{i}}$.

The edge ideal of a vertex-weighted digraph was first introduced by Gimenez et al [28]. Let $D=(V,E,w)$ be a vertex-weighted digraph with the vertex set $V=\{x_{1},\ldots,x_{n}\}$. We consider the polynomial ring $S=k[x_{1},\dots,x_{n}]$ in $n$ variables over a field $k$. The edge ideal of $D$, denoted by $I(D)$, is the ideal of $S$ given by

[TABLE]

Edge ideals of weighted digraphs arose in the theory of Reed-Muller codes as initial ideals of vanishing ideals of projective spaces over finite fields [42, 46]. If a vertex $x_{i}$ of $D$ is a source (i.e., has only arrows leaving $x_{i}$) we shall always assume $w_{i}=1$ because in this case the definition of $I(D)$ does not depend on the weight of $x_{i}$. If $w_{j}=1$ for all $j$, then $I(D)$ is the edge ideal of underlying graph $G(D)$ of $D$. Many researchers have tried to compute $a,b$ and $t_{0}$ such that $\mbox{reg}\,(I(G(D))^{t})=at+b$ for all $t\geq t_{0}$ for the edge ideals of some special families of graphs (see [1, 2, 5, 6, 9, 10, 43, 48]). Some authors have studied bounds for $\text{depth}\,(S/I(G(D))^{t})$ or $\text{pd}\,(I(G(D))^{t})$ for edge ideals of some special graphs (see [32, 33, 26, 44]). In [55], the first three authors derive some exact formulas for the projective dimension and regularity of the edge ideals of vertex-weighted rooted forests and oriented cycles. To the best of our knowledge, there is no result about the projective dimension and the regularity of $I(D)^{t}$ for a vertex-weighted digraph.

In this article, we are interested in algebraic properties corresponding to the projective dimension and the regularity of $I(D)^{t}$ for some vertex-weighted oriented graphs. By using the approaches of Betti splitting and polarization, we derive some exact formulas for the projective dimension and the regularity of powers of edge ideals of some directed graphs. The results are as follows:

Theorem 1.1**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted rooted forest such that $w(x)\geq 2$ if $d(x)\neq 1$. Then

[TABLE]

where $w=\mbox{max}\,\{w(x)\mid x\in V(D)\}$.

Theorem 1.2**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted rooted forest such that $w(x)\geq 2$ if $d(x)\neq 1$. Then

[TABLE]

Our paper is organized as follows. In section $2$, we recall some definitions and basic facts used in the following sections. In section $3$, we provide some exact formulas for the regularity of the powers of the edge ideals of vertex-weighted line graphs. Meanwhile, we give some examples to show the regularity of the powers of the edge ideals of vertex-weighted oriented line graphs are related to direction selection and the assumption that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped. In section $4$, we give some exact formulas for the projective dimension and the regularity of the powers of the edge ideals of vertex-weighted rooted forests. Moreover, we also give some examples to show the projective dimension and the regularity of the powers of the edge ideals of vertex-weighted oriented rooted forests are related to direction selection and the assumption that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.

For all unexplained terminology and additional information, we refer to [37] (for the theory of digraphs), [12] (for graph theory), and [13, 33] (for the theory of edge ideals of graphs and monomial ideals). We greatfully acknowledge the use of the computer algebra system CoCoA ([15]) for our experiments.

2. Preliminaries

In this section, we gather together the needed definitions and basic facts, which will be used throughout this paper. However, for more details, we refer the reader to [4, 6, 12, 21, 27, 29, 33, 36, 37, 42, 47, 53, 55].

A directed graph or digraph $D$ consists of a finite set $V(D)$ of vertices, together with a collection $E(D)$ of ordered pairs of distinct points called edges or arrows. If $\{u,v\}\in E(D)$ is an edge, we write $uv$ for $\{u,v\}$, which is denoted to be the directed edge where the direction is from $u$ to $v$ and $u$ (resp. $v$) is called the starting point (resp. the ending point). Given any digraph $D$, we can associate a graph $G$ on the same vertex set simply by replacing each arrow by an edge with the same ends. This graph is called the underlying graph of $D$, denoted by $G(D)$. Conversely, any graph $G$ can be regarded as a digraph, by replacing each of its edges by just one of the two oppositely oriented arrows with the same ends. Such a digraph is called an orientation of $G$. An orientation of a simple graph is referred to as an simple oriented graph.

Every concept that is valid for graphs automatically applies to digraphs too. For example, the degree of a vertex $x$ in a digraph $D$, denoted $d(x)$, is simply the degree of $x$ in $G(D)$. Likewise, a digraph is said to be connected if its underlying graph is connected. An oriented path or oriented cycle is an orientation of a path or cycle in which each vertex dominates its successor in the sequence. An oriented acyclic graph is a simple digraph without oriented cycles. An oriented tree or polytree is a oriented acyclic graph formed by orienting the edges of undirected acyclic graphs. A rooted tree is an oriented tree in which all edges are oriented either away from or towards the root. Unless specifically stated, a rooted tree in this article is an oriented tree in which all edges are oriented away from the root. An oriented forest is a disjoint union of oriented trees. A rooted forest is a disjoint union of rooted trees.

A vertex-weighted oriented graph is a triplet $D=(V(D),E(D),w)$, where $V(D)$ is the vertex set, $E(D)$ is the edge set and $w$ is a weight function $w:V(D)\rightarrow\mathbb{N}^{+}$, where $N^{+}=\{1,2,\ldots\}$. Some times for short we denote the vertex set $V(D)$ and edge set $E(D)$ by $V$ and $E$ respectively. The weight of $x_{i}\in V$ is $w(x_{i})$, denoted by $w_{i}$ or $w_{x_{i}}$.. Given a vertex-weighted oriented graph $D=(V,E,w)$ with the vertex set $V=\{x_{1},\ldots,x_{n}\}$, we may consider the polynomial ring $S=k[x_{1},\dots,x_{n}]$ in $n$ variables over a field $k$. The edge ideal of $D$, denoted by $I(D)$, is the ideal of $S$ given by

[TABLE]

If a vertex $x_{i}$ of $D$ is a source (i.e., has only arrows leaving $x_{i}$) we shall always assume $w_{i}=1$ because in this case the definition of $I(D)$ does not depend on the weight of $x_{i}$.

For any homogeneous ideal $I$ of the polynomial ring $S=k[x_{1},\dots,x_{n}]$, there exists a graded minimal finite free resolution

$0\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p,j}(I)}\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p-1,j}(I)}\rightarrow\cdots\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{0,j}(I)}\rightarrow I\rightarrow 0,$ where the maps are exact, $p\leq n$, and $S(-j)$ is the $S$-module obtained by shifting the degrees of $S$ by $j$. The number $\beta_{i,j}(I)$, the $(i,j)$-th graded Betti number of $I$, is an invariant of $I$ that equals the number of minimal generators of degree $j$ in the $i$th syzygy module of $I$. Of particular interests are the following invariants which measure the size of the minimal graded free resolution of $I$. The projective dimension of $I$, denoted pd $(I)$, is defined to be

[TABLE]

The regularity of $I$, denoted $\mbox{reg}\,(I)$, is defined by

[TABLE]

We now derive some formulas for $\mbox{pd}\,(I)$ and $\mbox{reg}\,(I)$ in some special cases by using some tools developed in [27].

Definition 2.1**.**

Let $I$ be a monomial ideal, and suppose that there exist monomial ideals $J$ and $K$ such that $\mathcal{G}(I)$ is the disjoint union of $\mathcal{G}(J)$ and $\mathcal{G}(K)$, where $\mathcal{G}(I)$ denotes the unique minimal set of monomial generators of $I$. Then $I=J+K$ is a Betti splitting if

[TABLE]

where $\beta_{i-1,j}(J\cap K)=0\hskip 5.69054pt\mbox{if}\hskip 5.69054pti=0$.

This formula was first obtained for the total Betti numbers by Eliahou and Kervaire [23] and extended to the graded case by Fatabbi [25]. In [27], the authors describe some sufficient conditions for an ideal $I$ to have a Betti splitting. We need the following lemma.

Lemma 2.2**.**

([27, Corollary 2.7]). Suppose that $I=J+K$ where $\mathcal{G}(J)$ contains all the generators of $I$ divisible by some variable $x_{i}$ and $\mathcal{G}(K)$ is a nonempty set containing the remaining generators of $I$. If $J$ has a linear resolution, then $I=J+K$ is a Betti splitting. Hence

[TABLE]

When $I$ is a Betti splitting ideal, Definition 2.1 implies the following results:

Corollary 2.3**.**

If $I=J+K$ is a Betti splitting ideal, then

• (1)

$\mbox{reg}\,(I)=\mbox{max}\{\mbox{reg}\,(J),\mbox{reg}\,(K),\mbox{reg}\,(J\cap K)-1\}$,

• (2)

$\mbox{pd}\,(I)=\mbox{max}\{\mbox{pd}\,(J),\mbox{pd}\,(K),\mbox{pd}\,(J\cap K)+1\}$.

We need the following Lemma:

Lemma 2.4**.**

([29, Lemma 2.2 and Lemma 3.2 ]) Let $S_{1}=k[x_{1},\dots,x_{m}]$, $S_{2}=k[x_{m+1},\dots,x_{n}]$ and $S=k[x_{1},\dots,x_{n}]$ be three polynomial rings, $I\subseteq S_{1}$ and $J\subseteq S_{2}$ be two proper non-zero homogeneous ideals. Then

• (1)

$\mbox{reg}\,(S/(I+J))=\mbox{reg}\,(S_{1}/I)+\mbox{reg}\,(S_{2}/J)$,

• (2)

$\mbox{pd}\,(S/(I+J))=\mbox{pd}\,(S_{1}/I)+\mbox{pd}\,(S_{2}/J)$.

Let $\mathcal{G}(I)$ denote the minimal set of generators of a monomial ideal $I\subset S$ and let $u\in S$ be a monomial, we set $\mbox{supp}(u)=\{x_{i}:x_{i}|u\}$. If $\mathcal{G}(I)=\{u_{1},\ldots,u_{m}\}$, we set $\mbox{supp}(I)=\bigcup\limits_{i=1}^{m}\mbox{supp}(u_{i})$. The following lemma is well known.

Lemma 2.5**.**

Let $I,J=(u)$ be two monomial ideals such that $\mbox{supp}\,(u)\cap\mbox{supp}\,(I)=\emptyset$. If the degree of $u$ is $d$. Then

• (1)

$\mbox{reg}\,(J)=d$,

• (2)

$\mbox{reg}\,(JI)=\mbox{reg}\,(I)+d$.

Definition 2.6**.**

Suppose that $u=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ is a monomial in $S$. We define the polarization of $u$ to be the squarefree monomial

[TABLE]

in the polynomial ring $S^{\mathcal{P}}=k[x_{ij}\mid 1\leq i\leq n,1\leq j\leq a_{i}]$. If $I\subset S$ is a monomial ideal with $\mathcal{G}(I)=\{u_{1},\ldots,u_{m}\}$, the polarization of $I$, denoted by $I^{\mathcal{P}}$, is defined as:

[TABLE]

which is a squarefree monomial ideal in the polynomial ring $S^{\mathcal{P}}$.

Here is an example of how polarization works.

Example 2.7**.**

Let $I(D)=(x_{1}x_{2}^{3},x_{2}x_{3},x_{3}x_{4}^{2},x_{4}x_{5}^{5})$ be the edge ideal of a vertex-weighted rooted tree $D$, then the polarization of $I(D)$ is the ideal $I(D)^{\mathcal{P}}=(x_{11}x_{21}x_{22}x_{23},\\ x_{21}x_{31},x_{31}x_{41}x_{42},x_{41}x_{51}x_{52}x_{53}x_{54}x_{55})$.

A monomial ideal $I$ and its polarization $I^{\mathcal{P}}$ share many homological and algebraic properties. The following is a very useful property of polarization.

Lemma 2.8**.**

([33, Corollary 1.6.3]) Let $I\subset S$ be a monomial ideal and $I^{\mathcal{P}}\subset S^{\mathcal{P}}$ its polarization. Then

• (1)

$\beta_{ij}(I)=\beta_{ij}(I^{\mathcal{P}})$* for all $i$ and $j$,*

• (2)

$\mbox{reg}\,(I)=\mbox{reg}\,(I^{\mathcal{P}})$,

• (2)

$\mbox{pd}\,(I)=\mbox{pd}\,(I^{\mathcal{P}})$.

The following lemma can be used for computing the projective dimension and the regularity of an ideal.

Lemma 2.9**.**

([31, Lemma 1.1 and Lemma 1.2]) Let  $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$   be a short exact sequence of finitely generated graded $S$-modules. Then

• (1)

$\mbox{pd}\,(B)=\mbox{pd}\,(A)$* if $\mbox{pd}\,(A)\geq\mbox{pd}\,(C)$,*

• (2)

$\mbox{reg}\,(B)\leq\mbox{max}\,\{\mbox{reg}\,(A),\mbox{reg}\,(C)\}$,

• (3)

$\mbox{reg}\,(B)=\mbox{reg}\,(C)$* if $\mbox{reg}\,(C)\geq\mbox{reg}\,(A)$,*

• (4)

$\mbox{reg}\,(B)=\mbox{reg}\,(A)$* if $\mbox{reg}\,(A)>\mbox{reg}\,(C)+1$,*

• (5)

$\mbox{reg}\,(C)=\mbox{reg}\,(A)-1$* if $\mbox{reg}\,(A)>\mbox{reg}\,(B)$,*

• (6)

$\mbox{reg}\,(C)=\mbox{reg}\,(B)$* if $\mbox{reg}\,(A)<\mbox{reg}\,(B)$.*

3. Regularity of powers of edge ideals of vertex-weighted oriented line graphs

In this section, by using the approaches of Betti splitting and polarization, we will provide some formulas for the regularity of the powers of the edge ideals of vertex-weighted oriented line graphs. We also give some examples to show the assumptions that $w(x)\geq 2$ if $d(x)\neq 1$ in oriented line graph cannot be dropped. We shall start with the following two lemmas.

Lemma 3.1**.**

([31, Lemma 1.3]) Let $R$ be the polynomial ring over a field and let $I$ be a proper homogeneous ideal in $R$. Then

• (1)

$\mbox{pd}\,(I)=\mbox{pd}\,(R/I)-1$,

• (2)

$\mbox{reg}\,(I)=\mbox{reg}\,(R/I)+1$.

Lemma 3.2**.**

Let $R=k[x_{1},\ldots,x_{m}]$ be the polynomial ring over a field $k$ and $I$ a proper homogeneous ideal in $R$. Let $y$ be another variable and $S=R[y]$. Then

• (1)

$\mbox{pd}\,(S/IS)=\mbox{pd}\,(R/IR)+1$,

• (2)

$\mbox{reg}\,(S/IS)=\mbox{reg}\,(R/IR)$.

Proof.

The results follow from $S/IS\cong(R/IR)[y]\cong R/IR\otimes_{k}k[y]$.

The following theorem generalizes Lemma 4.4 of [10].

Theorem 3.3**.**

Let $u_{1},u_{2},\ldots,u_{r}$ be a regular sequence of homogeneous polynomials in $S$ with $deg\,(u_{i})=d_{i}$ for $i=1,\ldots,r$. Let $I=(u_{1},u_{2},\ldots,u_{r})$ be an ideal. Then

[TABLE]

where $w=max\{d_{i}\mid 1\leq i\leq r\}$.

Proof.

We apply induction on $r$ and $t$. For $r=1$, the statements are clear for all $t\geq 1$. If $r\geq 2$, it is obvious that $\mbox{reg}\,(I)=\sum\limits_{i=1}^{r}d_{i}-(r-1)$ by Lemmas 2.4 and 3.1. Therefore, we may suppose that $r,t\geq 2$. For convenience, let’s assume that $w=d_{r}$ and $J=(u_{1},\ldots,u_{r-1})$. Then

[TABLE]

Hence there exists a surjection:

[TABLE]

Since $u_{r}$ is a regular element of $S/J$, the kernel of $\phi$ is $u_{r}J^{t}$. Thus we have the short exact sequence

[TABLE]

By induction hypotheses on $t$ and $r$, we obtain that

[TABLE]

and

[TABLE]

where $w^{\prime}=\mbox{max}\,\{d_{i}\mid 1\leq i\leq r-1\}$.

Notice that $w\geq w^{\prime}$, this implies $\mbox{reg}\,(I^{t-1}(-w))\geq\mbox{reg}\,(J^{t})$. It follows that

[TABLE]

Using the fact $u_{r}I^{t-1}\cap J^{t}=u_{r}J^{t}$, Lemma 2.5 (2) and induction hypothesis on $r$, we have

[TABLE]

Let $\alpha=\mbox{reg}\,(I^{t-1}(-w)\oplus J^{t})$ and $\beta=\mbox{reg}\,(u_{r}I^{t-1}\cap J^{t})$, then

[TABLE]

If $w^{\prime}=w$, then $\beta-\alpha=1$. By Lemma 2.9 (5), we have

[TABLE]

If $w^{\prime}\leq w-2$, or $w^{\prime}=w-1$ and $t\geq 3$, then $\beta<\alpha$. Thus by Lemma 2.9 (6), we have

[TABLE]

Therefore it is enough to prove this conclusion only for $t=2$ and $w=w^{\prime}+1$. In this case, $w\geq 2$. Set $d_{r-1}=w^{\prime}$ and $K=(u_{1},u_{2},\ldots,u_{r-2},u_{r})$, we have $I^{2}=(K+(u_{r-1}))I=u_{r-1}I+K^{2}$. Thus there exists the exact sequence

[TABLE]

Using the fact $w=w^{\prime}+1$, $u_{r-1}I\cap K^{2}=u_{r-1}K^{2}$ and induction hypothesis, we have

[TABLE]

[TABLE]

and

[TABLE]

It follows that $\mbox{reg}\,(u_{r-1}K^{2})>\mbox{max}\,\{\mbox{reg}(I(-w^{\prime})),\mbox{reg}\,(K^{2})\}$ because of $w\geq 2$, thus the result follows from Lemma 2.9 (5).

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented graph. For $T\subset V(D)$, we define the induced vertex-weighted subgraph $H=(V(H),E(H),w)$ of $D$ to be the vertex-weighted oriented graph such that $V(H)=T$, $uv\in E(H)$ if and only if $uv\in E(D)$ and for any $u,v\in V(H)$. For any $u\in V(H)$ and $u$ is not a source in $H$, its weight in $H$ equals to the weight of $u$ in $D$, otherwise, its weight in $H$ equals to $1$. For $P\subset V(D)$, we denote $D\setminus P$ the induced subgraph of $D$ obtained by removing the vertices in $P$ and the edges incident to these vertices. If $P=\{x\}$ consists of a single element, then we write $D\setminus x$ for $D\setminus\{x\}$. For $W\subseteq E(D)$, we define $D\setminus W$ to be the subgraph of $D$ with all edges in $W$ deleted (but its vertices remained). When $W=\{e\}$ consists of a single edge, we write $D\setminus e$ instead of $D\setminus\{e\}$.

Let $x\in V(D)$, then we call $N_{D}^{+}(x)=\{y:(x,y)\in E(D)\}$ and $N_{D}^{-}(x)=\{y:(y,x)\in E(D)\}$ to be the out-neighbourhood and in-neighbourhood of $x$, respectively. The neighbourhood of $x$ is the set $N_{D}(x)=N_{D}^{+}(x)\cup N_{D}^{-}(x)$.

The following lemmas are needed to facilitate calculating the projective dimension and the regularity of powers through induction on the power.

Lemma 3.4**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented graph, let $z$ be a leaf with $N_{D}^{-}(z)=\{y\}$. Then

[TABLE]

Proof.

It’s clear that $(I(D\setminus z)^{t},z^{w_{z}})\subseteq(I(D)^{t},z^{w_{z}})$. If any monomial $f\in\mathcal{G}(I(D)^{t})\setminus\mathcal{G}(I(D\setminus z)^{t})$, then $z$ divides $f$. It follows that $yz^{w_{z}}$ also divides $f$ because of $N_{D}^{-}(z)=\{y\}$. This implies $f\in(z^{w_{z}})$.

Lemma 3.5**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented graph as in Lemma 3.4 such that $z$ is a leaf with $N_{D}^{-}(z)=\{y\}$. Then

[TABLE]

Proof.

If any monomial $f\in\mathcal{G}(I(D)^{t}:yz^{w_{z}})$, then $fyz^{w_{z}}\in I(D)^{t}$. We can write $fyz^{w_{z}}=e_{i1}e_{i2}\ldots e_{it}h$ for some monomial $h$, where $e_{ij}=x_{ij}y_{ij}^{w(y_{ij})}$ such that $x_{ij}y_{ij}\in E(D)$. If there exists some $j\in\{1,\ldots,t\}$ such that $z^{w_{z}}$ divides $e_{ij}$, then $e_{ij}=yz^{w_{z}}$ because of $N_{D}^{-}(z)=\{y\}$. This implies that $f\in I(D)^{t-1}$. If $z^{w_{z}}$ does not divide $e_{ij}$ for all $j\in\{1,\ldots,t\}$, then $z^{w_{z}}$ divides $h$. Thus $f\in I(D)^{t-1}$.

Lemma 3.6**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted oriented graph as in Lemma 3.4 such that $z$ is a leaf with $N_{D}^{-}(z)=\{y\}$. Then

[TABLE]

Proof.

It’s obvious that $((I(D\setminus y)^{t}:z^{w_{z}}),y)=(I(D\setminus y)^{t},y)$ by similar arguments as the proof of Lemma 3.5. Now assume that $f\in\mathcal{G}((I(D)^{t}:z^{w_{z}}),y)$ and $y$ does not divide $f$, then $f\in(I(D)^{t}:z^{w_{z}})$. It follows that $fz^{w_{z}}\in I(D\setminus y)^{t}$. This means that $f\in(I(D\setminus y)^{t}:z^{w_{z}})$. Hence $f\in((I(D\setminus y)^{t}:z^{w_{z}}),y)$.

Lemma 3.7**.**

Let $n\geq 3$ be an integer and $P_{n}$ a vertex-weighted oriented line graph with edge set $E(P_{n})=\{x_{1}x_{2},x_{2}x_{3},\ldots,x_{n-1}x_{n}\}$, its edge ideal $I(P_{n})=(x_{1}x_{2}^{w_{2}},x_{2}x_{3}^{w_{3}},\\ \ldots,x_{n-1}x_{n}^{w_{n}})$ such that $w_{i}\geq 2$ for any $2\leq i\leq n-1$. Let $I_{n}$ be an ideal with the generator set $\mathcal{G}(I_{n})=\mathcal{G}(I(P_{n})^{2})\setminus\{x_{1}^{2}x_{2}^{2w_{2}},x_{2}^{2}x_{3}^{2w_{3}},\ldots,x_{n-1}^{2}x_{n}^{2w_{n}}\}$. Then

[TABLE]

where $w=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n\}$.

Proof.

We apply induction on $n$. The case $n=3$ is clear. Assume that $n\geq 4$. Consider the short exact sequences

[TABLE]

and

[TABLE]

Following the same arguments as Lemmas 3.4 $\sim$ 3.6, we have $(I_{n},x_{n}^{w_{n}})=(I_{n-1},x_{n}^{w_{n}})$, $((I_{n}:x_{n}^{w_{n}}),x_{n-1})=(I_{n-2},x_{n-1})$ and $(I_{n}:x_{n-1}x_{n}^{w_{n}})=I(P_{n-1}).$

By induction hypothesis on $n$, Lemma 2.4 (1), Lemma 3.1 and [54, Theorem 3.5], we obtain

[TABLE]

where the last inequality holds because of $w^{\prime}\leq w$, where $w^{\prime}=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n-1\}$,

[TABLE]

and

[TABLE]

where the last inequality holds because of $w_{n-1}\geq 2$ and $w^{\prime\prime}\leq w$, here $w^{\prime\prime}=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n-2\}$.

By Lemma 3.1 (2) and using Lemma 2.9 (2) on the short exact sequence (2) and (4), (5), we have

[TABLE]

Again by using Lemma 2.9 (2) on the short exact sequence (1), and (3), (6), we have

[TABLE]

Now we are ready to present the main result of this section

Theorem 3.8**.**

Let $n\geq 2$ be an integer and $P_{n}$ a vertex-weighted oriented line graph, let $I(P_{n})=(x_{1}x_{2}^{w_{2}},x_{2}x_{3}^{w_{3}},\ldots,x_{n-1}x_{n}^{w_{n}})$ the edge ideal of $P_{n}$ with $w_{i}\geq 2$ for any $2\leq i\leq n-1$. Then

[TABLE]

where $w=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n\}$.

Proof.

It is sufficient to show $\mbox{reg}\,(S/I(P_{n})^{t})=\sum\limits_{x\in V(P_{n})}w(x)-|E(P_{n})|+1+(t-1)(w+1)-1$ for any $t\geq 1$ by Lemma 3.1 (2). We use induction on $n$ and $t$. The case $n=2$ is obvious. Assume that $n\geq 3$. For $t=1$, the statement is true by [54, Theorem 3.5]. Now we consider the case $t=2$. Let $J$ be the polarization of $I(P_{n})^{2}$, then

[TABLE]

For $1\leq i\leq n-1$, we let $K_{i}=(x_{n-i,1}x_{n-i,2}\!\!\!\!\!\prod\limits_{j=1}^{2w_{n-i+1}}\!\!\!\!\!x_{n-i+1,j})$,

[TABLE]

where $\Widehat{x_{n-i,1}x_{n-i,2}\!\!\!\!\prod\limits_{j=1}^{2w_{n-i+1}}\!\!\!\!x_{n-i+1,j}}$ denotes the element $x_{n-i,1}x_{n-i,2}\!\!\!\!\prod\limits_{j=1}^{2w_{n-i+1}}\!\!\!\!x_{n-i+1,j}$ being omitted from $J_{i}$. Then we have

[TABLE]

where $J_{0}=J$, $L_{1}=(x_{11}\!\!\prod\limits_{j=1}^{w_{2}}x_{2j},x_{21}\!\!\prod\limits_{j=1}^{w_{3}}x_{3j},\ldots,x_{n-3,1}\!\!\!\!\prod\limits_{j=1}^{w_{n-2}}\!\!\!\!x_{n-2,j},x_{n-2,1}\!\!\!\!\!\!\prod\limits_{j=3}^{w_{n-1}+1}\!\!\!\!x_{n-1,j})$,

$L_{2}=(x_{11}\!\!\prod\limits_{j=1}^{w_{2}}\!\!x_{2j},x_{21}\!\!\prod\limits_{j=1}^{w_{3}}\!\!x_{3j},\ldots,x_{n-4,1}\!\!\prod\limits_{j=1}^{w_{n-3}}\!\!x_{n-3,j},x_{n-3,1}\!\!\!\!\!\prod\limits_{j=3}^{w_{n-2}+1}\!\!\!\!\!x_{n-2,j},\!\!\prod\limits_{j=1}^{w_{n}}\!\!x_{nj})$,

$L_{i}=(x_{11}\prod\limits_{j=1}^{w_{2}}x_{2j},x_{21}\!\!\prod\limits_{j=1}^{w_{3}}\!\!x_{3j},\ldots,x_{n-i-2,1}\!\!\!\!\!\prod\limits_{j=1}^{w_{n-i-1}}\!\!\!\!\!x_{n-i-1,j},x_{n-i-1,1}\!\!\!\!\!\prod\limits_{j=3}^{w_{n-i}+1}\!\!\!\!\!x_{n-i,j},\prod\limits_{j=1}^{w_{n-i+2}}\!\!x_{n-i+2,j},\\ x_{n-i+2,1}\!\!\prod\limits_{j=1}^{w_{n-i+3}}\!\!\!x_{n-i+3,j},\ldots,x_{n-1,1}\!\prod\limits_{j=1}^{w_{n}}\!x_{nj})\ \text{for}\ 3\leq i\leq n-1$ and variables that appear in $K_{i}$ and $L_{i}$ are different for $1\leq i\leq n-1$.

By Lemma 2.4, [54, Theorem 3.5] and the proof of [54, Theorem 4.1], we obtain

[TABLE]

[TABLE]

In brief, for $1\leq i\leq n-1$, we have

[TABLE]

Notice that all $K_{i}$ have linear resolutions for $1\leq i\leq n-1$, it follows that $J_{i-1}=J_{i}+K_{i}$ is a Betti splitting and $\mbox{reg}\,(K_{i})=2w_{n-i+1}+2$. By Lemma 2.2, we obtain, for $1\leq i\leq n-1$,

[TABLE]

By repeated use of the above equalities (2) and Lemma 2.8 (2), we can obtain

[TABLE]

Notice that the ideal $I_{n}$ in Lemma 3.7 is the polarization $J_{n-1}^{\mathcal{P}}$ of $J_{n-1}$. By Lemmas 2.8 (2) and 3.7, we have

[TABLE]

Thus, by equalities (1), (3) and (4), we obtain

[TABLE]

where the second equality holds because of $\mbox{reg}\,(J_{n-1})\leq\sum\limits_{i=1}^{n}w_{i}-(n-1)+1+(w+1)$ and $w=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n\}$.

Finally we assume that $t\geq 3$. Consider the short exact sequences

[TABLE]

[TABLE]

Notice that $(I(P_{n})^{t},x_{n}^{w_{n}})=(I(P_{n-1})^{t},x_{n}^{w_{n}})$, $(I(P_{n})^{t}:x_{n-1}x_{n}^{w_{n}})=I(P_{n})^{t-1}$ and $((I(P_{n})^{t}:x_{n}^{w_{n}}),x_{n-1})=(I(P_{n-2})^{t},x_{n-1})$ by Lemmas 3.4 $\sim$ 3.6. Thus, by induction hypotheses on $n$ and $t$, and Lemma 2.4 (1) and Lemma 3.1 (2), we get

[TABLE]

where $w^{\prime}=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n-1\}$ and $S^{\prime}=k[x_{1},\ldots,x_{n-1}]$;

[TABLE]

and

[TABLE]

where the second equality holds by Lemma 3.2 (2), here $w^{\prime\prime}=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n-2\}$ and $S^{\prime\prime}=k[x_{1},x_{2},\ldots,x_{n-2}]$. Let

[TABLE]

and

[TABLE]

then, by equality (8) and (9), we get

[TABLE]

If $w=w^{\prime}$, then $\alpha\geq\mbox{max}\,\{\beta,\gamma\}$ by comparing (7), (8) and (9). Using Lemma 2.9 (3) on the short exact sequence (6), we obtain

[TABLE]

Again using Lemma 2.9 (3) on the short exact sequence (5), we get

[TABLE]

If $w>w^{\prime}$, then $w=w_{n}$ and $w>w^{\prime\prime}$. This implies $\beta-\gamma>1$ because of $t\geq 3$. Using Lemma 2.9 (4) on the exact sequence (6), we obtain

[TABLE]

By comparing equalities (7) and (10), we get

[TABLE]

because of $t\geq 3$. Therefore, by using Lemma 2.9 (4) on the exact sequence (5), and equality (10), we have

[TABLE]

The proof is completed.

The following example shows that the assumption in Theorem 3.8 that $w_{i}\geq 2$ for any $2\leq i\leq n-1$ cannot be dropped.

Example 3.9**.**

Let $I(D)=(x_{1}x_{2}^{5},x_{2}x_{3},x_{3}x_{4}^{8})$ be the edge ideal of vertex-weighted oriented line graph $D=(V(D),E(D),w)$ with $w_{1}=w_{3}=1$, $w_{2}=5$ and $w_{4}=8$. By using CoCoA, we get $\mbox{reg}\,(I(D)^{2})=18$. But we have $\mbox{reg}\,(I(D)^{2})=\sum\limits_{i=1}^{4}w_{i}-|E(P_{n})|+1+(w_{4}+1)=22$ by Theorem 3.8.

The following example shows that the regularity of the powers of the edge ideals of vertex-weighted oriented line graphs are related to direction selection in Theorem 3.8.

Example 3.10**.**

Let $I(D)=(x_{1}x_{2}^{5},x_{3}x_{2}^{5},x_{3}x_{4}^{8},x_{5}x_{4}^{8},x_{5}x_{6}^{2})$ be the edge ideal of a vertex-weighted oriented line graph $D=(V(D),E(D),w)$ with $w_{1}=w_{3}=w_{5}=1$, $w_{2}=5$, $w_{4}=8$ and $w_{6}=2$. By using CoCoA, we obtain $\mbox{reg}\,(I(D)^{2})=17$. But we have $\mbox{reg}\,(I(D)^{2})=\sum\limits_{i=1}^{6}w_{i}-|E(P_{n})|+1+(w_{4}+1)=23$ by Theorem 3.8.

4. projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests

In this section, we will give some formulas for the projective dimension and the regularity of the powers of the edge ideals of vertex-weighted rooted forests. We shall start from oriented star graph.

Theorem 4.1**.**

Let $D=(V(D),E(D),w)$ be a weighted oriented star graph. If its edge set $E(D)$ is one of the following three cases $\{x_{1}x_{2},x_{1}x_{3},\ldots,x_{1}x_{n}\}$, $\{x_{2}x_{1},x_{3}x_{1},\\ \ldots,x_{n}x_{1}\}$, and $\{x_{1}x_{2},x_{2}x_{3},\ldots,x_{2}x_{n}\}$ where $w_{2}\geq 2$. Then

[TABLE]

where $w=\mbox{max}\,\{w_{i}\mid 1\leq i\leq n\}$.

Proof.

The cases $E(D)=\{x_{1}x_{2},x_{1}x_{3},\ldots,x_{1}x_{n}\}$ and $E(D)=\{x_{2}x_{1},x_{3}x_{1},\ldots,x_{n}x_{1}\}$ can be shown by similar arguments, we only consider the case $E(D)=\{x_{1}x_{2},x_{1}x_{3},\\ \ldots,x_{1}x_{n}\}$, then

[TABLE]

From Lemmas 2.5 (2), 3.1 (2), 2.4 (1) and Theorem 3.3, it follows that

[TABLE]

If $E(D)=\{x_{1}x_{2},x_{2}x_{3},\ldots,x_{2}x_{n}\}$, then

[TABLE]

It is sufficient to show $\mbox{reg}\,(\frac{S}{I(D)^{t}})=\sum\limits_{i=1}^{n}w_{i}-|E(D)|+1+(t-1)(w+1)-1$ for any $t\geq 1$ by Lemma 3.1 (2). We prove this statement by induction on $n$ and $t$. The cases $n=2$ and $n=3$ follow from Theorem 3.8. Assume that $n\geq 4$. The case $t=1$ follows from [54, Theorem 3.1]. Now assume that $t\geq 2$. Without lose of generality, we may assume $w_{n}\leq w_{n-1}$. Consider the following short exact sequences

[TABLE]

and

[TABLE]

Note that $(I(D)^{t},x_{n}^{w_{n}})=(I(D\setminus x_{n})^{t},x_{n}^{w_{n}})$ by Lemma 3.4. Thus, by Lemma 2.4 (1), Lemma 3.1 (2) and induction hypothesis on $n$, we have

[TABLE]

where the last equality holds because of $w^{\prime}:=\mbox{max}\{w_{i}\mid 1\leq i\leq n-1\}=w$ and $S^{\prime}=k[x_{1},x_{2},\ldots,x_{n-1}]$.

Since $(I(D)^{t}:x_{2}x_{n}^{w_{n}})=I(D)^{t-1}$ by Lemma 3.5, by using induction hypothesis on $t$ and Lemma 3.1 (2), we get

[TABLE]

Notice that $((I(D)^{t}:x_{n}^{w_{n}}),x_{2})=(I(D\setminus x_{2})^{t},x_{2})=(x_{2})$ by Lemma 3.6, it follows that

[TABLE]

By comparing the equalities (4) and (5), we get

[TABLE]

By the above inequality, the exact sequence (2) and Lemma 2.9 (4), we have

[TABLE]

where the last equality holds from (4). By comparing with the equalities (3) and (6), we have

[TABLE]

Therefore, by the exact sequence (1), Lemma 2.9 (3) and Lemma 3.1 (2), we obtain that

[TABLE]

where the last equality holds from (3). The proof is completed.

Now, we prove the main results of this section.

Theorem 4.2**.**

Proof.

We prove this statement by induction on $|V(D)|$ and $t$. The case $|V(D)|=2$ follows from Theorem 3.8. Assume that $|V(D)|\geq 3$. The case $t=1$ follows from [54, Theorem 3.5]. Now assume that $t\geq 2$. If $D$ is an oriented line graph, it follows from Theorem 3.8. Otherwise, there are at least two leaves in $D$. We may suppose both $x$ and $z$ are leaves of $D$ such that $w_{z}\leq w_{x}$ and $N_{D}^{-}(z)=\{y\}$. If $D$ is an oriented star graph, then this statement holds by Theorem 4.1.

If there exists a connected component $D_{1}$ of $D$, which is a line graph with only one edge $\{y,z\}$. Then $I(D)=(I(D\setminus z),yz^{w_{z}})$ and

[TABLE]

Thus there exists a surjection:

[TABLE]

Since $yz^{w_{z}}$ is a regular element of $S/I(D\setminus z)$, the kernel of $\phi$ is $yz^{w_{z}}I(D\setminus z)^{t}$. Thus we have the short exact sequence

[TABLE]

By induction hypotheses on $t$ and $|V(D)|$, we obtain that

[TABLE]

and

[TABLE]

where the second equality holds in formula for $\mbox{reg}\,(I(D\setminus z)^{t})$ because of $w_{z}\leq w_{x}$ and $w^{\prime}=\mbox{max}\,\{w(x)\mid x\in V(D\setminus z)\}$,

[TABLE]

It follows that

[TABLE]

Thus the result follows from Lemma 2.9 (5).

Otherwise, we will show $\mbox{reg}\,(\frac{S}{I(D)^{t}})=\!\!\!\sum\limits_{x\in V(D)}\!\!\!w(x)-|E(D)|+1+(t-1)(w+1)-1$ for any $t\geq 1$. Thus the conclusion follows from Lemma 3.1 (2).

We consider the following short exact sequences

[TABLE]

and

[TABLE]

Notice that $(I(D)^{t},z^{w_{z}})=(I(D\setminus z)^{t},z^{w_{z}})$, $(I(D)^{t}:yz^{w_{z}})=I(D)^{t-1}$ and $((I(D)^{t}:z^{w_{z}}),y)=(I(D\setminus y)^{t},y)$ by Lemmas 3.2 $\sim$ 3.5. Thus by Lemma 2.4 (1) and induction hypotheses on $|V(D)|$ and $t$, we obtain that

[TABLE]

where $w^{\prime}=\mbox{max}\,\{w(x)\mid x\in V(D\setminus z)\}$ and $S^{\prime}$ is a polynomial ring over a field with the variable set $V(D\setminus z)$,

[TABLE]

Let $N_{D}(y)=N_{D}^{-}(y)\cup N_{D}^{+}(y)$, where $N_{D}^{-}(y)=\{x_{1}\}$, $N_{D}^{+}(y)=\{z,x_{2},\ldots,x_{\ell},x_{{\ell}+1},\\ \ldots,x_{m}\}$ such that $\{z,x_{2},\ldots,x_{\ell}\}$ being leaves of $D$ and $\{x_{\ell+1},\ldots,x_{m}\}$ not being leaves of $D$ or $N_{D}(y)=N_{D}^{+}(y)=\{z,x_{1},x_{2},\ldots,x_{\ell},x_{{\ell}+1},\ldots,x_{m}\}$ such that $\{z,x_{1},\ldots,x_{\ell}\}$ being leaves of $D$ and $\{x_{\ell+1},\ldots,x_{m}\}$ not being leaves of $D$. The second case means that $w_{y}=1$. Thus by Lemma 2.4 (1) and induction hypothesis on $|V(D)|$, we obtain

(1) If $N_{D}^{-}(y)=\{x_{1}\}$, then

[TABLE]

(2) If $N_{D}^{-}(y)=\emptyset$, then

[TABLE]

where the last equality holds because $\{x_{\ell+1},\ldots,x_{m}\}$ being roots of $D\setminus y$, here $w^{\prime\prime}=\mbox{max}\,\{w(x)\mid x\in V(D\setminus y)\}$ and $S^{\prime\prime}$ is a polynomial ring over a field with the variable set $V(D\setminus y)$.

Using Lemma 2.9 (2) on the short exact sequence (2), we have

[TABLE]

By comparing the equalities (3), (4) (5) and (6), we get

[TABLE]

Again using Lemma 2.9 (3) on the short exact sequence (1), we obtain that

[TABLE]

The proof is completed.

As a consequence of Theorem 4.2, we have

Corollary 4.3**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted rooted forest as in Theorem 4.2. Then

[TABLE]

where $w=\mbox{max}\,\{w(x)\mid x\in V(D)\}$.

Proof.

This is a direct consequence of the above theorem and [54, Theorem 3.5].

Theorem 4.4**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted rooted forest such that $w(x)\geq 2$ if $d(x)\neq 1$. Then

[TABLE]

Proof.

We prove this statement by induction on $|E(D)|$ and $t$. The case $|E(D)|=1$ is clear. Assume that $|E(D)|\geq 2$. The case $t=1$ follows from [54, Theorem 3.3]. Now assume that $t\geq 2$. We may suppose $z$ is a leaf of $D$ and $N_{D}^{-}(z)=\{y\}$. we will show $\mbox{pd}\,(\frac{S}{I(D)^{t}})=|E(D)|$ for any $t\geq 1$. Thus the conclusion follows from Lemma 3.1 (1).

Consider the following short exact sequences

[TABLE]

and

[TABLE]

Notice that $(I(D)^{t},z^{w_{z}})=(I(D\setminus z)^{t},z^{w_{z}})$, $(I(D)^{t}:yz^{w_{z}})=I(D)^{t-1}$ and $((I(D)^{t}:z^{w_{z}}),y)=(I(D\setminus y)^{t},y)$ by Lemmas 3.4 $\sim$ 3.6. We set $N_{D}(y)=\{z,x_{1},x_{2},\ldots,x_{m}\}$, this implies $|E(D\setminus y)|=|E(D)|-(m+1)$. Thus by Lemma 2.4 (2) and induction hypotheses on $|E(D)|$ and $t$, we obtain

[TABLE]

where $S^{\prime}$ is the polynomial ring with variable set $V(D\setminus z)$,

[TABLE]

and

[TABLE]

where $S^{\prime\prime}$ is the polynomial ring with variable set $V(D\setminus y)$.

Using Lemma 2.9 (1) on the short exact sequence (1), (2) and equalities (3), (4) and (5), we get

[TABLE]

An immediate consequence of the above theorem is the following corollary.

Corollary 4.5**.**

Let $D=(V(D),E(D),w)$ be a vertex-weighted rooted forest as in Theorem 4.4. Then $\mbox{depth}\,(I(D))=n-|E(D)|+1$.

Proof.

By Auslander-Buchsbaum formula (see Theorem 1.3.3 of [13]), it follows that

[TABLE]

The following example shows that the assumption in Theorems 4.2 and 4.4 that $D$ is a vertex-weighted rooted forest such that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.

Example 4.6**.**

Let $I(D)=(x_{1}x_{2}^{2},x_{2}x_{3},x_{3}x_{4}^{2},x_{5}x_{6}^{2},x_{6}x_{7},x_{7}x_{8}^{2})$ be the edge ideal of weighted oriented forest $D=(V(D),E(D),w)$ with $w_{1}=w_{3}=w_{5}=w_{7}=1$ and $w_{2}=w_{4}=w_{6}=w_{8}=2$. By using CoCoA, we obtain $\mbox{pd}\,(I(D)^{2})=4$ and $\mbox{reg}\,(I(D)^{2})=8$. But we have $\mbox{pd}\,(I(D)^{2})=|E(D)|-1=5$ by Theorem 4.4 and $\mbox{reg}\,(I(D)^{2})=\sum\limits_{i=1}^{8}w_{i}-|E(D)|+1+(w_{2}+1)=10$ by Theorem 4.2.

The following example shows that the projective dimension of the powers of the edge ideals of vertex-weighted oriented forest are related to direction selection in Theorem 4.4.

Example 4.7**.**

Let $I(D)=(x_{1}x_{2}^{5},x_{3}x_{2}^{5},x_{3}x_{4}^{8},x_{5}x_{4}^{8},x_{5}x_{6}^{2})$ be the edge ideal of vertex-weighted oriented line graph $D=(V(D),E(D),w)$ with $w_{1}=w_{3}=w_{5}=1$, $w_{2}=5$, $w_{4}=8$ and $w_{6}=2$. By using CoCoA, we obtain $\mbox{pd}\,(I(D)^{2})=3$. But we get $\mbox{pd}\,(I(D)^{2})=|E(D)|-1=4$ by Theorem 4.4.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (No.11271275) and by foundation of the Priority Academic Program Development of Jiangsu Higher Education Institutions.