# Symbolic powers in weighted oriented graphs

Mousumi Mandal∗ and Dipak Kumar Pradhan  Department of Mathematics, Indian
Institute of Technology Kharagpur, 721302, India<EMAIL_ADDRESS>Department of Mathematics, Indian Institute of Technology Kharagpur, 721302,
India<EMAIL_ADDRESS>
###### Abstract.

Let $D$ be a weighted oriented graph with the underlying graph $G$ when
vertices with non-trivial weights are sinks and $I(D),I(G)$ be the edge ideals
corresponding to $D$ and $G,$ respectively. We give an explicit description of
the symbolic powers of $I(D)$ using the concept of strong vertex covers. We
show that the ordinary and symbolic powers of $I(D)$ and $I(G)$ behave in a
similar way. We provide a description for symbolic powers and Waldschmidt
constant of $I(D)$ for certain classes of weighted oriented graphs. When $D$
is a weighted oriented odd cycle, we compute
$\operatorname{reg}(I(D)^{(s)}/I(D)^{s})$ and prove
$\operatorname{reg}I(D)^{(s)}\leq\operatorname{reg}I(D)^{s}$ and show that
equality holds when there is only one vertex with non-trivial weight.

Keywords: Weighted oriented graph, edge ideal, symbolic power, Waldschmidt
constant, Castelnuovo-Mumford regularity.

AMS Classification 2010: 05C22, 13D02, 13D45, 05C25, 05C38, 05E40

##  1\. Introduction

A directed graph or digraph $D=(V(D),E(D))$ consists of a finite set $V(D)$ of
vertices together with a prescribed collection $E(D)$ of ordered pairs of
distinct vertices called edges or arrows. If $(u,v)\in E(D),$ then we call it
a directed edge where the direction is from $u$ to $v$ and $u$ (respectively
$v$) is called the initial vertex (respectively the terminal vertex). An
oriented graph is a directed graph without multiple edges or loops. In other
words, an oriented graph $D$ is a simple graph $G$ together with an
orientation of its edges. An oriented graph $D$ is called vertex weighted
oriented if it is equipped with a weight function
$w:V(D)\longrightarrow\mathbb{N}$. In fact, a vertex weighted oriented graph
$D$ is a triplet $D=(V(D),E(D),w)$ where $V(D)$ and $E(D)$ are the vertex set
and edge set, respectively, and the weight of a vertex $x_{i}\in V(D)$ is
$w(x_{i})$ denoted by $w_{i}$ or $w_{x_{i}}.$ We set $V^{+}(D)=\\{x\in
V(D)~{}|~{}w(x)\neq 1\\}$ and it is denoted by $V^{+}$. If
$V(D)=\\{x_{1},\ldots,x_{n}\\},$ we can regard each vertex $x_{i}$ as a
variable and consider the polynomial ring $R=k[x_{1},\ldots,x_{n}]$ over a
field $k$. Then the edge ideal of $D$ is defined as

$I(D)=(x_{i}x_{j}^{w_{j}}~{}|~{}(x_{i},x_{j})\in E(D)).$

The underlying graph of $D$ is the simple graph $G$ whose vertex set is
$V(G)=V(D)$ and whose edge set is $E(G)=\\{\\{u,v\\}~{}|~{}(u,v)\in E(D)\\}.$
The edge ideal of $G$ is $I(G)=(uv~{}|~{}\\{u,v\\}\in E(G))\subseteq
R=k[x_{1},\ldots,x_{n}].$ 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
$w(x)=1$ for all $x\in V,$ then $I(D)$ recovers the usual edge ideal of the
underlying graph $G$. Here, we shall always mean that a weighted oriented
graph is the same as a vertex weighted oriented graph. The interest in edge
ideals of weighted digraphs comes from coding theory, especially, in the study
of Reed-Muller types codes. The edge ideal of a vertex weighted digraph
appears as initial ideals of vanishing ideals of certain projective spaces
over finite fields [15].
Algebraic invariants and properties like Cohen-Macaulayness and unmixedness of
edge ideals of weighted oriented graphs have been studied in [13] and [16].
Recently in [1] and [20], regularity and projective dimension of the edge
ideals of some classes of weighted oriented graphs have been studied. In [16],
Y. Pitones et al. studied some properties of weighted oriented graphs when
$V(D)$ is a strong vertex cover and the elements of $V^{+}$ are sinks. They
described the irreducible decomposition of $I(D)$ for any weighted oriented
graph $D$ using the concept of strong vertex cover. In the geometrical
context, symbolic powers are important since they capture all the polynomials
that vanish with a given multiplicity. For any homogeneous ideal $I\subset R,$
its $m$-th symbolic power is defined as
$I^{(m)}=\displaystyle{\bigcap_{p\in\operatorname{Ass}I}(I^{m}R_{p}\cap R)}$.
In [19], a classical result of Zariski and Samuel states that $I^{(m)}=I^{m}$
for all $m\geq 1$ if $I$ is generated by a regular sequence, or equivalently,
a complete intersection. Ideals that have the property $I^{(m)}=I^{m}$ for all
$m\geq 1$ are called normally torsion free because their Rees algebra is
normal. In particular, in [7] Gitler et al. showed that a squarefree monomial
ideal is normally torsion free if and only if the corresponding hypergraph
satisfies the max-flow min-cut property. Symbolic powers of codimension $2$
Cohen-Macaulay ideals have been studied recently in [6]. In the last decade
the comparison between symbolic and regular powers of ideals is studied not
only for graphs but also for ideal defining a set of points both in projective
and multiprojective spaces [2, 8, 9]. Recently symbolic powers and invariants
of the edge ideals of simple graphs have been studied in [4, 11, 14]. But
nothing is known about the symbolic powers of edge ideals of weighted oriented
graphs.
In general, computing the symbolic powers of ideals is a difficult problem.
Symbolic powers of square free monomial ideals have been studied via the
concept of minimal vertex cover for edge ideals of graphs. Here we study the
symbolic powers of some classes of non square free monomial ideals through the
edge ideals of weighted oriented graphs using the concept of strong vertex
cover. We also compare the regularity of the symbolic and ordinary powers of
the edge ideals of weighted oriented odd cycles when the elements of $V^{+}$
are sinks.
In Section 2, we recall all the definitions and results which will be required
for the rest of the paper. In Section 3, we provide a relation between
ordinary and symbolic powers of edge ideals of some classes of weighted
naturally oriented graphs in Corollary 3.2. In Theorem 3.4, we show that the
$m$-th symbolic power of edge ideal of a weighted oriented graph, when the
elements of $V^{+}$ are sinks, is the intersection of the $m$-th powers of
irreducible ideals associated to the strong vertex covers. In Theorem 3.7, we
prove that the ordinary and symbolic powers of edge ideal of a weighted
oriented graph behaves in the similar way as in the edge ideal of its
underlying graph in case of elements of $V^{+}$ are sinks.
We also give an explicit description of symbolic powers and Waldschmidt
constants of edge ideals of weighted oriented graphs when the elements of
$V^{+}$ are sinks and underlying graphs are unicyclic graph with a unique odd
cycle, complete graph and clique sum of two odd cycles of same length joining
at a common vertex (see Proposition 3.9, Proposition 3.10 and Proposition
3.12, respectively). In Section 4, in Theorem 4.3, we provide an inequality
between the regularity of ordinary and symbolic powers of edge ideal of a
weighted oriented odd cycle when the elements of $V^{+}$ are sinks and in
Corollary 4.5, we prove the equality when $V^{+}$ contains a single element.

##  2\. Preliminaries

In this section, we recall some definitions and results regarding a weighted
oriented graph $D$ and its underlying graph $G.$ A vertex cover of $G$ is a
subset $V^{\prime}\subseteq V(G)$ such that for all $e\in E(G),$ $e\cap
V^{\prime}\neq\phi.$ A minimal vertex cover is a vertex cover which is minimal
with respect to inclusion.
The next two lemmas describe the symbolic powers of the edge ideals in terms
of minimal vertex covers.

######  Lemma 2.1.

[2, Lemma 2.5] Let $G$ be a graph on vertices $\\{x_{1},\ldots,x_{n}\\},$
$I=I(G)\subseteq k[x_{1},\ldots,x_{n}]$ be the edge ideal of $G$ and
$V_{1},\ldots,V_{r}$ be the minimal vertex covers of $G.$ Let $P_{j}$ be the
monomial prime ideal generated by the variables in $V_{j}.$ Then

$I=P_{1}\cap\cdots\cap P_{r}$

and

$I^{(m)}=P_{1}^{m}\cap\cdots\cap P_{r}^{m}.$

######  Lemma 2.2.

[2, Lemma 2.6] Let $I\subseteq S$ be a squarefree monomial ideal with minimal
primary decomposition $I=P_{1}\cap\cdots\cap
P_{r}~{}~{}with~{}~{}P_{j}=(x_{j_{1}},\ldots,x_{js_{j}})$ for $j=1,\ldots,r$.
Then ${x_{1}^{a_{1}}}\cdots{x_{n}^{a_{n}}}\in I^{(m)}\mbox{ if and only if
}a_{j_{1}}+\cdots+a_{j_{s_{j}}}\geq m$ for $j=1,\ldots,r$.

Below we recall some definitions and results for the weighted oriented graph
$D.$

######  Definition 2.3.

A vertex cover $C$ of $D$ is a subset of $V(D)$ such that if $(x,y)\in E(D),$
then $x\in C$ or $y\in C.$ A vertex cover $C$ of $D$ is minimal if each proper
subset of $C$ is not a vertex cover of $D.$ We set $(C)$ to be the ideal
generated by the variables in $C.$

######  Remark 2.4.

[16, Remark 2] $C$ is a minimal vertex cover of $D$ if and only if $C$ is a
minimal vertex cover of $G.$

######  Definition 2.5.

Let $x$ be a vertex of a weighted oriented graph $D,$ then the sets
$N_{D}^{+}(x)=\\{y:(x,y)\in E(D)\\}$ and $N_{D}^{-}(x)=\\{y:(y,x)\in E(D)\\}$
are called the out-neighbourhood and the in-neighbourhood of $x$,
respectively. Moreover, the neighbourhood of $x$ is the set
$N_{D}(x)=N_{D}^{+}(x)\cup N_{D}^{-}(x).$ Define $\deg_{D}(x)=|N_{D}(x)|$ for
$x\in V(D)$. A vertex $x\in V(D)$ is called a source vertex if
$N_{D}(x)=N_{D}^{+}(x).$ A vertex $x\in V(D)$ is called a sink vertex if
$N_{D}(x)=N_{D}^{-}(x).$

######  Definition 2.6.

[16, Definition 4] Let $C$ be a vertex cover of a weighted oriented graph $D,$
we define
$L_{1}(C)=\\{x\in C~{}|~{}N_{D}^{+}(x)\cap C^{c}\neq\phi\\},$
$L_{2}(C)=\\{x\in C~{}|~{}x\notin L_{1}(C)~{}\mbox{and}~{}N_{D}^{-}(x)\cap
C^{c}\neq\phi\\}$ and
$L_{3}(C)=C\setminus(L_{1}(C)\cup L_{2}(C))$
where $C^{c}$ is the complement of $C,$ i.e., $C^{c}=V(D)\setminus C.$

######  Lemma 2.7.

[16, Proposition 6] Let $C$ be a vertex cover of $D.$ Then $L_{3}(C)=\phi$ if
and only if $C$ is a minimal vertex cover of $D.$

######  Definition 2.8.

[16, Definition 7] A vertex cover $C$ of $D$ is strong if for each $x\in
L_{3}(C)$ there is $(y,x)\in E(D)$ such that $y\in L_{2}(C)\cup L_{3}(C)$ with
$y\in V^{+}$ (i.e., $w(y)\neq 1$).

######  Definition 2.9.

[16, Definition 32] A weighted oriented graph $D$ has the minimal-strong
property if each strong vertex cover is a minimal vertex cover.

The following lemma gives a class of weighted oriented graphs which satisfy
the minimal-strong property.

######  Lemma 2.10.

[16, Lemma 47] If the elements of $V^{+}$ are sinks, then $D$ has the minimal-
strong property.

The next lemma describes the irreducible decomposition of the edge ideal of
weighted oriented graph $D$ in terms of irreducible ideals associated with the
strong vertex covers of $D$.

######  Lemma 2.11.

[16, Theorem 25, Remark 26] Let $D$ be a weighted oriented graph and
$C_{1},\ldots,C_{s}$ are the strong vertex covers of $D,$ then the irredundant
irreducible decomposition of $I(D)$ is

$I(D)=I_{C_{1}}\cap\cdots\cap I_{C_{s}}$

where each $I_{C_{i}}=(L_{1}(C_{i})\cup\\{x_{j}^{w(x_{j})}~{}|~{}x_{j}\in
L_{2}(C_{i})\cup L_{3}(C_{i})\\}),$
$\operatorname{rad}(I_{C_{i}})=P_{i}=(C_{i})$.

######  Corollary 2.12.

[16, Remark 26] Let $D$ be a weighted oriented graph. Then $P$ is an
associated prime of $I(D)$ if and only if $P=(C)$ for some strong vertex cover
$C$ of $D.$

######  Notation 2.13.

The degree of a monomial $m\in k[x_{1},\ldots,x_{n}]$ is denoted by $\deg(m).$

Now we recall the definition of Castelnuovo-Mumford regularity using local
cohomology. Let $M$ be a non zero module over the polynomial ring $R$. If
$H^{i}_{\mathfrak{m}}(M)$ denotes the $i$-th local cohomology module of the
$R$-module $M$ with support on the maximal ideal $\mathfrak{m},$ we set

$a_{i}(M)=\max\\{j\in\mathbb{Z}:[H^{i}_{\mathfrak{m}}(M)]_{j}\neq 0\\}$

(or $a_{i}(M)=-\infty$ if $H^{i}_{\mathfrak{m}}(M)=0$) where
$[H^{i}_{\mathfrak{m}}(M)]_{j}$ denotes the $j$-th graded component of the
$i$-th local cohomology module of $M.$ Then the Castelnuovo-Mumford regularity
of $M$ is defined as

$\operatorname{reg}M=\max_{0\leq i\leq\dim M}\\{a_{i}(M)+i\\}.$

One of the important asymptotic invariants for symbolic powers is resgurence
which was introduced by Bocci and Harbourne [3] to answer the ideal
containment problem. In general, computation of resurgence of an ideal is
difficult. In [3], one lower bound of resurgence of an ideal is given in terms
of another invariant known as Waldschmidt constant of $I.$ Let $\alpha(I)$
$=\min\\{d~{}|~{}I_{d}\neq 0\\},$ i.e., $\alpha(I)$ is the smallest degree of
a nonzero element in $I.$ Then the Waldschmidt constant of $I$ is defined as

$\widehat{\alpha}{(I)}=\displaystyle{\lim_{s\rightarrow\infty}{}\frac{\alpha{(I^{(s)})}}{s}}.$

##  3\. Symbolic Powers in Oriented Graphs

We provide an explicit description of the symbolic powers for edge ideals of
weighted oriented graphs when the elements of $V^{+}$ are sinks. In the next
lemma, we give a sufficient condition for the equality of the symbolic and
ordinary powers of the edge ideals for some classes of weighted oriented
graphs.

######  Lemma 3.1.

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D.$ If $V(D)$ is
a strong vertex cover of $D,$ then $I^{(k)}=I^{k}$ for all $k.$

###### Proof.

Since $V(D)$ is the strong vertex cover of $D,$ by Corollary 2.12, $P=(V(D))$
is an associated prime of $I(D).$ Here $P$ is the unique maximal associated
prime in $\operatorname{Ass}(I).$ Thus by [5, Lemma 3.3], we have
$I^{(k)}=I^{k}$ for all $k.$ ∎

A cycle is naturally oriented if all edges of cycle are oriented in clockwise
direction. In a naturally oriented unicyclic graph, the cycle is naturally
oriented and each edge of the tree connected with the cycle is oriented away
from the cycle.

######  Corollary 3.2.

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D.$ If there are
weighted naturally oriented unicyclic graphs $D_{1},\ldots,D_{s}$ of $D$ such
that $V(D_{1}),\ldots,V(D_{s})$ is a partition of $V(D)$ and $w(x)\neq 1$ if
$\deg_{D_{i}}(x)>1$ for each $i,$ then $I^{(k)}=I^{k}$ for all $k.$

###### Proof.

Since there are weighted naturally oriented unicyclic graphs
$D_{1},\ldots,D_{s}$ of $D$ such that $V(D_{1}),\ldots,V(D_{s})$ is a
partition of $V(D)$ and $w(x)\neq 1$ if $\deg_{D_{i}}(x)>1$ for each $i,$ by
[16, Proposition 15], $V(D)$ is a strong vertex cover of $D.$ Therefore by
Lemma 3.1, we have $I^{(k)}=I^{k}$ for all $k.$ ∎

######  Corollary 3.3.

Let $I=I(D)$ be the edge ideal of a weighted oriented complete graph $D$ on
the vertex set $\\{x_{1},\ldots,x_{n}\\}$ where the cycle
$C_{n}=(x_{1},\ldots,x_{n})$ is naturally oriented and the diagonals are
oriented in any direction such that $w(x)\neq 1$ for any vertex $x.$ Then
$I^{(k)}=I^{k}$ for all $k.$

###### Proof.

Let $D_{1}$ be the weighted naturally oriented cycle whose underlying graph is
the cycle $C_{n}=(x_{1},\ldots,x_{n}).$ Here $V(D_{1})=V(D)$ and $w(x)\neq 1$
for all $x\in V(D_{1}).$ Then by Corollary 3.2, we have $I^{(k)}=I^{k}$ for
all $k.$ ∎

In the next theorem, we describe the symbolic powers for edge ideals of
weighted oriented graphs when the elements of $V^{+}$ are sinks, in terms of
irreducible ideals associated to the strong vertex covers. For the remainder
of this section, we set $V^{+}\neq\phi.$

######  Theorem 3.4.

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$ where the
elements of $V^{+}$ are sinks and $C_{1},\ldots,C_{s}$ be the strong vertex
covers of $D.$ Let $w_{j}=w(x_{j})$ if $x_{j}\in V^{+}.$ Then the irredundant
irreducible decomposition of $I$ is

$I=I_{C_{1}}\cap\cdots\cap I_{C_{s}}$

and

$I^{(m)}={I_{C_{1}}}^{m}\cap\cdots\cap{I_{C_{s}}}^{m}$

where each $I_{C_{i}}=(\\{x_{j}~{}|~{}x_{j}\in C_{i}\setminus
V^{+}\\}\cup\\{x_{j}^{w_{j}}~{}|~{}x_{j}\in C_{i}\cap V^{+}\\}),$
$\operatorname{rad}(I_{C_{i}})=P_{i}=(C_{i})$ and $P_{i}$’s are minimal primes
of $I.$

###### Proof.

Here the elements of $V^{+}$ are sinks. By Lemma 2.11,
$I=I_{C_{1}}\cap\cdots\cap I_{C_{s}}$ be the irredundant irreducible
decomposition of $I$ where each
$I_{C_{i}}=(L_{1}(C_{i})\cup\\{x_{j}^{w_{j}}~{}|~{}x_{j}\in L_{2}(C_{i})\cup
L_{3}(C_{i})\\})$, $\operatorname{rad}(I_{C_{i}})=P_{i}=(C_{i})$ and $P_{i}$’s
are associated primes of $I.$ By Lemma 2.10, $C_{1},\ldots,C_{s}$ are minimal
vertex covers of $D$. Thus by Lemma 2.7, we have $L_{3}(C_{i})=\phi$ for
$i=1,\ldots,s.$ For a fixed $i,$ if $x\in C_{i}$ is a sink vertex, then
$N_{D}^{+}(x)\cap C_{i}^{c}=\phi,$ which implies $x\notin L_{1}(C_{i}).$ Thus
$L_{2}(C_{i})=C_{i}\cap V^{+}$ and $L_{1}(C_{i})=C_{i}\setminus V^{+}.$
Therefore $I_{C_{i}}=(\\{x_{j}~{}|~{}x_{j}\in C_{i}\setminus
V^{+}\\}\cup\\{x_{j}^{w_{j}}~{}|~{}x_{j}\in C_{i}\cap V^{+}\\}).$ Since
$C_{i}$’s are minimal vertex covers of $D$, the $P_{i}$’s are minimal primes
of $I.$ Hence by [5, Lemma 3.7], we have
$I^{(m)}={I_{C_{1}}}^{m}\cap\cdots\cap{I_{C_{s}}}^{m}.$ ∎

######  Proposition 3.5.

Let $I$ be the edge ideal of a weighted oriented graph $D$ where the elements
of $V^{+}$ are sinks with irredundant irreducible decomposition
$I=I_{C_{1}}\cap\cdots\cap I_{C_{s}}$ where
$\operatorname{rad}(I_{C_{i}})=P_{i}=(C_{i})=(x_{i_{1}},\ldots,x_{i_{r_{i}}})$
for $i=1,\ldots,s$ and $C_{i}$’s are the strong vertex covers of $D.$ Let
$w_{j}=w(x_{j})$ if $x_{j}\in V^{+}.$ Then every minimal generator of
$I^{(m)}$ is of the form $(\prod x_{j}^{a_{j}}|x_{j}\notin
V^{+})(\prod{x_{j}^{w_{j}}}^{a_{j}}|x_{j}\in V^{+})$ for some $a_{j}$’s where
$a_{i_{1}}+\cdots+a_{i_{r_{i}}}\geq m$ for $i=1,\ldots,s.$

###### Proof.

Here each $I_{C_{i}}$ is of the form $I_{C_{i}}=(\\{x_{j}~{}|~{}x_{j}\in
C_{i}\setminus V^{+}\\}\cup\\{x_{j}^{w_{j}}~{}|~{}x_{j}\in C_{i}\cap
V^{+}\\})$ and
$\operatorname{rad}(I_{C_{i}})=P_{i}=(C_{i})=(x_{i_{1}},\ldots,x_{i_{r_{i}}}).$
By Theorem 3.4, we have
$I^{(m)}={I_{C_{1}}}^{m}\cap\cdots\cap{I_{C_{s}}}^{m}.$ Let $t=(\prod
x_{j}^{a_{j}}|x_{j}\notin V^{+})(\prod{x_{j}^{w_{j}}}^{a_{j}}|x_{j}\in
V^{+})(\prod x_{j}^{b_{j}}|x_{j}\in V^{+},b_{j}<w_{j})$ for some $a_{j}$’s and
$b_{j}$’s. Then $t\in I^{(m)}$ if and only if $t\in{I_{{C_{i}}}}^{m}$ for
$i=1,\ldots,s.$ This is possible if and only if there exists at least one
generator $f_{i}\in{I_{C_{i}}}^{m}$ such that $f_{i}$ divides $t$ for
$i=1,\ldots,s$ which is equivalent to the condition
$a_{i_{1}}+\cdots+a_{i_{r_{i}}}\geq m$ for $i=1,\ldots,s.$
From the structure of $I^{(m)}={I_{C_{1}}}^{m}\cap\cdots\cap{I_{C_{s}}}^{m},$
it is clear that every minimal generator of $I^{(m)}$ is of the form $(\prod
x_{j}^{a_{j}}|x_{j}\notin V^{+})(\prod{x_{j}^{w_{j}}}^{a_{j}}|x_{j}\in V^{+})$
for some $a_{j}$’s where $a_{i_{1}}+\cdots+a_{i_{r_{i}}}\geq m$ for
$i=1,\ldots,s.$ ∎

Now we relate the symbolic powers of edge ideals of weighted oriented graphs
with the edge ideals of underlying graphs and show that both of them behave in
a similar way when the elements of $V^{+}$ are sinks.

######  Notation 3.6.

Let $D$ be a weighted oriented graph where the elements of $V^{+}$ are sinks
and $w_{j}=w(x_{j})$ if $x_{j}\in V^{+}.$ Here $R=k[x_{1},\ldots,x_{n}]=$
$\displaystyle{{\bigoplus_{d=0}^{\infty}}R_{d}}$ is the standard graded
polynomial ring. Consider the map

$\displaystyle\Phi:R\longrightarrow R~{}\mbox{where}~{}x_{j}\longrightarrow
x_{j}~{}\mbox{if}~{}x_{j}\notin V^{+}\mbox{and}~{}x_{j}\longrightarrow
x_{j}^{w_{j}}~{}\mbox{if}~{}x_{j}\in V^{+}.$

Here $\Phi$ is an injective homomorphism of k-algebras. We use this definition
of $\Phi$ for the following theorem.

######  Theorem 3.7.

Let $D$ be a weighted oriented graph where the elements of $V^{+}$ are sinks
and $G$ be the underlying graph of $D.$ Let $I$ and $\tilde{I}$ be the edge
ideals of $G$ and $D,$ respectively. Then $\Phi(I^{k})=\tilde{I}^{k}$ and
$\Phi(I^{(k)})=\tilde{I}^{(k)}$ for all $k\geq 1.$

###### Proof.

Let $V(G)=V(D)=\\{x_{1},\ldots,x_{n}\\}.$ Let $C_{1},\ldots,C_{s}$ are the
strong vertex covers of $D$ and by Lemma 2.10, they are minimal vertex covers
of $D$. By Remark 2.4, $C_{1},\ldots,C_{s}$ are also minimal vertex covers of
$G.$ Thus by Lemma 2.1, the minimal primary decomposition of $I$ is
$I=P_{1}\cap\cdots\cap P_{s}$ where
$P_{i}=(C_{i})=(x_{i_{1}},\ldots,x_{i_{r_{i}}})$ for $i=1,\ldots,s.$ By Lemma
2.11, $\tilde{I}={\tilde{I}}_{C_{1}}\cap\cdots\cap{\tilde{I}}_{C_{s}}$ be the
irredundant irreducible decomposition of $I$ where each
${\tilde{I}}_{C_{i}}=(\\{x_{j}~{}|~{}x_{j}\in C_{i}\setminus
V^{+}\\}\cup\\{x_{j}^{w_{j}}~{}|~{}x_{j}\in C_{i}\cap V^{+}\\}),$
$w_{j}=w(x_{j})$ if $x_{j}\in V^{+},$
$\operatorname{rad}({\tilde{I}}_{C_{i}})=P_{i}=(C_{i})$ and $P_{i}$’s are
minimal primes of $\tilde{I}.$ Here $\Phi(I)={\tilde{I}}.$ Then
$\Phi(I^{k})=[\Phi(I)]^{k}={\tilde{I}^{k}}$ for all $k\geq 1.$
Next we want to prove that $\Phi(I^{(k)})={\tilde{I}^{(k)}}.$ By Lemma 2.1,
$I^{(k)}=P_{1}^{k}\cap\cdots\cap P_{s}^{k}$ and by Theorem 3.4, we have
${\tilde{I}}^{(k)}={{\tilde{I}}_{C_{1}}}^{k}\cap\cdots\cap{{\tilde{I}}_{C_{s}}}^{k}.$
Here $\Phi(P_{i})={\tilde{I}}_{C_{i}}$ for $i=1,\ldots,s.$
Let $q$ be a minimal generator of $\Phi(I^{(k)}).$ Then there exists $p\in
I^{(k)}$ such that $q=\Phi(p).$ Since $p\in I^{(k)}=P_{1}^{k}\cap\cdots\cap
P_{s}^{k},$ then $p\in P_{i}^{k}$ for each $i=1,\ldots,s$ which implies that
$\Phi(p)\in\Phi({P_{i}}^{k})=[\Phi(P_{i})]^{k}={\tilde{I}_{C_{i}}^{k}}.$ Thus
$q=\Phi(p)\in{{\tilde{I}}_{C_{1}}}^{k}\cap\cdots\cap{{\tilde{I}}_{C_{s}}}^{k}={\tilde{I}}^{(k)}.$
Therefore $\Phi(I^{(k)})\subseteq{\tilde{I}^{(k)}}.$
Let $q$ be a minimal generator of ${\tilde{I}}^{(k)}.$ Then by Proposition
3.5, we have $q=(\prod x_{j}^{a_{j}}|x_{j}\notin
V^{+})(\prod{x_{j}^{w_{j}}}^{a_{j}}|x_{j}\in V^{+})$ for some $n$-tuple
$(a_{1},\ldots,a_{n})$ where $a_{i_{1}}+\cdots+a_{i_{r_{i}}}\geq k$ for
$i=1,\ldots,s.$ By letting $p:=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}},$ we see that
$\Phi(p)=q$ and by Lemma 2.2, $p\in I^{(k)}$ since
$a_{i_{1}}+\cdots+a_{i_{r_{i}}}\geq k$ for $i=1,\ldots,s$. Thus
$q=\Phi(p)\in\Phi(I^{(k)}).$ Therefore
${\tilde{I}^{(k)}}\subseteq\Phi(I^{(k)}).$ Hence
$\Phi(I^{(k)})={\tilde{I}^{(k)}}$ for all $k\geq 1.$ ∎

Next, we apply the above theorem to some specific classes of weighted oriented
graphs and give an explicit description of the symbolic powers of their edge
ideals. Then we compute the Waldschmidt constant.

######  Corollary 3.8.

Let $\tilde{I}$ be the edge ideal of a weighted oriented bipartite graph $D$
where the elements of $~{}V^{+}$ are sinks and let $G$ be the underlying graph
of $D$ which is a bipartite graph. Then ${\tilde{I}^{(k)}}={\tilde{I}^{k}}$
for all $k\geq 1.$

###### Proof.

Let $I$ be the edge ideal of $G$ and $\Phi$ is same as defined in Notation
3.6. Since $G$ is a bipartite graph, we have ${{I}^{(k)}}={{I}^{k}}$ for all
$k\geq 1$ by [17, Theorem 5.9]. Thus by Theorem 3.7, we have
$\Phi({{I}^{(k)}})=\Phi({{I}^{k}}),$ i.e., ${\tilde{I}^{(k)}}={\tilde{I}^{k}}$
for all $k\geq 1.$ ∎

In [2, Theorem 6.7], C. Bocci et al. found that the Waldschmidt constant of
edge ideal of an odd cycle $C_{2n+1}$ is $\displaystyle{\frac{2n+1}{n+1}}$. In
the next Proposition, we observe that the Waldschmidt constant of edge ideal
of a weighted oriented unicyclic graph depends on the position of the elements
of $V^{+}.$

######  Proposition 3.9.

Let $\tilde{I}$ be the edge ideal of a weighted oriented unicyclic graph $D$
where the elements of $V^{+}$ are sinks and let $G$ be the underlying graph of
$D$ with a unicycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $c=(\prod
x_{j}|x_{j}\in V(C_{2n+1})\setminus V^{+})(\prod{x_{j}^{w(x_{j})}}|x_{j}\in
V(C_{2n+1})\cap V^{+}).$

  1. (1)

Then ${\tilde{I}^{(k)}}={\tilde{I}^{k}}$ for $1\leq k\leq n$ and
${\tilde{I}^{(n+1)}}={\tilde{I}^{n+1}}+(c).$

  2. (2)

Let $s\in\mathbb{N}$ and write $s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and
$0\leq r\leq n.$ Then

$\tilde{I}^{(s)}=\displaystyle{\sum_{t=0}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}}.$

  3. (3)

If none of the vertices of $V(C_{2n+1})$ is in $V^{+}$ and some vertex of
$V(G)\setminus V(C_{2n+1})$ is in $V^{+},$ then $\alpha{(\tilde{I}^{(s)})}$ =
$2s-\Bigl{\lfloor}\dfrac{s}{n+1}\Bigr{\rfloor}\qquad$ for $s\in\mathbb{N}.$
In particular, the Waldschmidt constant of $\tilde{I}$ is given by

$\displaystyle\widehat{\alpha}{(\tilde{I})}=\frac{2n+1}{n+1}.$

  4. (4)

If at least one of the vertices of $V(C_{2n+1})$ is in $V^{+}$, then
$\alpha{(\tilde{I}^{(s)})}$ = $2s$ for $s\in\mathbb{N}.$
In particular, the Waldschmidt constant of $\tilde{I}$ is given by

$\displaystyle\widehat{\alpha}{(\tilde{I})}=2.$

###### Proof.

Let $I$ be the edge ideal of $G$ and $\Phi$ is same as defined in Notation
3.6.

  1. (1)

Since $G$ is a unicyclic graph with a unique odd cycle
$C_{2n+1}=(x_{1},\ldots,x_{2n+1}),$ by [11, Lemma 3.3] , we have
${{I}^{(k)}}={{I}^{k}}$ for $1\leq k\leq n$ and
${{I}^{(n+1)}}={{I}^{n+1}}+(x_{1}\cdots x_{2n+1}).$ Thus, by Theorem 3.7,
$\Phi({{I}^{(k)}})=\Phi({{I}^{k}}),$ i.e., ${\tilde{I}^{(k)}}={\tilde{I}^{k}}$
for $1\leq k\leq n$ and
$\Phi({{I}^{(n+1)}})=\Phi({{I}^{n+1}})+\Phi(x_{1}\cdots x_{2n+1}),$ i.e.,
${\tilde{I}^{(n+1)}}={\tilde{I}^{n+1}}+(c).$

  2. (2)

By [11, Lemma 3.4], we have
${I}^{(s)}=\displaystyle{\sum_{t=0}^{k}{I}^{s-t(n+1)}(x_{1}\cdots
x_{2n+1})^{t}.}$ Then by Theorem 3.7, applying $\Phi$ on both sides, we get
$\tilde{I}^{(s)}=\displaystyle{\sum_{t=0}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}.}$

  3. (3)

Assume that none of the vertices of $V(C_{2n+1})$ is in $V^{+}$. Thus by (1),
${\tilde{I}^{(n+1)}}={\tilde{I}^{n+1}}+(c)={\tilde{I}^{n+1}}+(x_{1}\cdots
x_{2n+1}).$ Since all the edges of cycle are without any element of $V^{+}$,
$\alpha(\tilde{I}^{s})=\alpha(I^{s})=2s$ for all $s.$ Here $\alpha(c)=2n+1,$
thus we have $\alpha(\tilde{I}^{(n+1)})=\alpha(I^{(n+1)})=2n+1.$ Since
${I}^{(s)}$ is generated by ${I}$ and ${I}^{(n+1)},$ $\tilde{I}^{(s)}$ is
generated by $\tilde{I}$ and $\tilde{I}^{(n+1)}$ which implies
$\alpha{(\tilde{I}^{(s)})}=\alpha{({I}^{(s)})}.$ Therefore by [11, Theorem
3.6], we have $\alpha{(\tilde{I}^{(s)})}$
$\displaystyle{=2s-\Bigl{\lfloor}\dfrac{s}{n+1}\Bigr{\rfloor}}.$ Hence
$\displaystyle{\widehat{\alpha}{(\tilde{I})}=2-\frac{1}{n+1}=\frac{2n+1}{n+1}}.$

  4. (4)

Assume that at least one of the vertices of $V(C_{2n+1})$ is in $V^{+}$. By
(1), we have
$\tilde{I}^{(s)}=\tilde{I}^{s}+\displaystyle{\sum_{t=1}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}.}$
Since $G$ contains a unique odd cycle, there is an edge of odd cycle without
any element of $V^{+}.$ Thus $\alpha(\tilde{I}^{s})=\alpha(I^{s})=2s$ for all
$s.$ Since the cycle $C_{2n+1}$ contains at least one element of $V^{+},$
$\alpha(c)\geq 2(n+1)$ and $\alpha(c^{t})\geq 2t(n+1).$ Also
$\alpha(\tilde{I}^{s-t(n+1)})=2s-2t(n+1)$ and
$\alpha({\tilde{I}}^{s-t(n+1)}(c)^{t})\geq 2s-2t(n+1)+2t(n+1)=2s$ for $1\leq
t\leq k.$ Since $\alpha(\tilde{I}^{s})=2s,$
$\alpha(\tilde{I}^{(s)})=\alpha(\tilde{I}^{s})=2s.$ Hence
$\widehat{\alpha}{(\tilde{I})}=2.$

∎

######  Proposition 3.10.

Let $\tilde{I}$ be the edge ideal of a weighted oriented complete graph $D$
where the elements of $V^{+}$ are sinks and let $G$ be the underlying graph of
$D$ on the vertices $\\{x_{1},\ldots,x_{n}\\}.$ Then

  1. (1)

For any $s\geq n,$ we have
$\displaystyle{\tilde{I}^{(s)}=\sum_{\begin{subarray}{c}(r_{1},\dots,r_{n-1})~{}and~{}\\\
s=r_{1}+2r_{2}+\dots+(n-1)r_{n-1}\end{subarray}}{\tilde{I}^{r_{1}}{{\tilde{I}}^{{(2)}^{r_{2}}}}\cdots{{\tilde{I}}^{{(n-1)}^{r_{n-1}}}}}}.$

  2. (2)

For any $s\in\mathbb{N}$, $\alpha{(\tilde{I}^{(s)})}$
$\displaystyle{=s+\Bigl{\lceil}\dfrac{s}{n-2}\Bigr{\rceil}.\qquad}$
In particular, the Waldschmidt constant of $\tilde{I}$ is given by

$\displaystyle\widehat{\alpha}{(\tilde{I})}=\frac{n-1}{n-2}.$

###### Proof.

Let $I$ be the edge ideal of $G$ and $\Phi$ is same as defined in Notation
3.6.

  1. (1)

By [4, Theorem 4.2] for any $s\geq n,$ we have

$I^{(s)}=\sum_{\begin{subarray}{c}(r_{1},\dots,r_{n-1})~{}and~{}\\\
s=r_{1}+2r_{2}+\dots+(n-1)r_{n-1}\end{subarray}}{I^{r_{1}}{{I^{(2)}}^{r_{2}}}\cdots{{I^{(n-1)}}^{r_{n-1}}}}.$

By Theorem 3.7, applying $\Phi$ on both sides we get

$\displaystyle{\tilde{I}^{(s)}=\sum_{\begin{subarray}{c}(r_{1},\dots,r_{n-1})~{}and~{}\\\
s=r_{1}+2r_{2}+\cdots+(n-1)r_{n-1}\end{subarray}}{\tilde{I}^{r_{1}}{{\tilde{I}}^{{(2)}^{r_{2}}}}\cdots{{\tilde{I}}^{{(n-1)}^{r_{n-1}}}}}}.$

  2. (2)

Since $D$ is a weighted oriented complete graph and all the elements of
$V^{+}$ are sinks, there is only one element in $V^{+}.$ Without loss of
generality, we may assume that the only element in $V^{+}$ is $x_{n}$ where
$w_{n}:=w(x_{n}).$ For $1\leq s\leq n-1$ by the definition of $I^{(s)},$ no
monomial of degree $s$ is in $I^{(s)}.$ Note that for $1\leq s\leq n-1,$
$x_{1}x_{2}\cdots x_{s+1}\in I^{(s)}.$ Then by Theorem 3.7,
$\Phi(x_{1}x_{2}\cdots x_{s+1})=x_{1}x_{2}\cdots x_{s+1}\in{\tilde{I}}^{(s)}$
for $1\leq s\leq n-2$ and $\Phi(x_{1}x_{2}\cdots
x_{n})=x_{1}x_{2}\cdots{x_{n}}^{w_{n}}\in{\tilde{I}}^{(n-1)}.$ Since degree of
any monomial can not be reduced under the map $\Phi,$ $x_{1}x_{2}\cdots
x_{s+1}$ is one of the least degree generator in both $I^{(s)}$ and
${\tilde{I}}^{(s)}$ for $1\leq s\leq n-2.$ Hence
$\alpha({\tilde{I}}^{(s)})=\alpha({{I}}^{(s)})=s+1$ for $1\leq s\leq n-2.$
Since $x_{1}\cdots x_{n}$ is the only minimal generator of degree $n$ in
$I^{(n-1)},$ there is no minimal generator of degree $n$ in
${\tilde{I}}^{(n-1)}$ as $\deg(x_{1}x_{2}\cdots{x_{n}}^{w_{n}})\geq n+1.$ By
the definition of $I^{(s)},$ $x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}}\in I^{(n-1)}.$
Then by Theorem 3.7,
$\Phi(x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}})=x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}}\in\tilde{I}^{(n-1)}.$
Since $\deg(x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}})=n+1,$
$x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}}$ is one of the least degree generator of
${\tilde{I}}^{(n-1)}$ and hence $\alpha({\tilde{I}}^{(n-1)})=n+1.$ Here
$x_{1}^{2}x_{2}^{2}\cdots{x_{n-1}}=(x_{1}x_{2}\cdots{x_{n-1}})(x_{1}x_{2})\in\tilde{I}^{(n-1)}$
where $x_{1}x_{2}\cdots{x_{n-1}}$ and $x_{1}x_{2}$ are one of the least degree
generators of ${\tilde{I}}^{(n-2)}$ and ${\tilde{I}}$ respectively. Therefore
one of the least degree generator of ${\tilde{I}}^{(n-1)}$ can be generated by
the least degree generators of ${\tilde{I}}^{(n-2)}$ and ${\tilde{I}}.$
By the above argument, one of the least degree generators of
${\tilde{I}}^{(s)}$ can be generated by the least degree generators of
${\tilde{I}},$ ${\tilde{I}^{(2)}},\ldots,{\tilde{I}}^{(n-2)}.$ Now from (1),
by assuming $r_{n-1}=0,$ it follows that for $s\geq n,$
$\alpha({\tilde{I}}^{(s)})=\min\\{2r_{1}+3r_{2}+\dots+(n-1)r_{n-2}\mid
s=r_{1}+2r_{2}+\dots+(n-2)r_{n-2}\\}.$ Now
$2r_{1}+3r_{2}+\dots+(n-1)r_{n-2}=s+r_{1}+r_{2}+\dots+r_{n-2}.$ Then it is
equivalent to find the minimum of ${r=r_{1}+r_{2}\dots+r_{n-2}}$ with the
condition $s=r_{1}+2r_{2}+\dots+(n-2)r_{n-2}.$ Write $s=k(n-2)+p$ for some
$k\in\mathbb{Z}$ and $0\leq p\leq n-3.$ Here the minimum value of $r$ will
occur for maximum value of $r_{n-2}$ and the maximum value of $r_{n-2}$ is
$k.$ So the minimal generating degree term will come from
${\tilde{I}}^{(n-2)^{k}}{\tilde{I}}^{(p)}.$ Thus

$\displaystyle\alpha({\tilde{I}}^{(s)})=\left\\{\begin{array}[]{ll}k(n-1)+(p+1)&\text{
if $p\neq 0,$}\\\ k(n-1)&\text{ if $p=0$ }\\\ \end{array}\right.$
$\displaystyle\alpha({\tilde{I}}^{(s)})=\left\\{\begin{array}[]{ll}s+1+\Bigl{\lfloor}\dfrac{s}{n-2}\Bigr{\rfloor}&\text{
if $p\neq 0,$}\\\ &\\\ s+\Bigl{\lfloor}\dfrac{s}{n-2}\Bigr{\rfloor}&\text{ if
$p=0$ }\\\ \end{array}\right.$

Therefore
$\displaystyle{\alpha{({\tilde{I}}^{(s)})}=s+\Bigl{\lceil}\dfrac{s}{n-2}\Bigr{\rceil}}.$
Also
$\displaystyle{\frac{s}{n-2}\leq\Bigl{\lceil}\dfrac{s}{n-2}\Bigr{\rceil}\leq\frac{s}{n-2}+1}.$
Hence

$\displaystyle{\widehat{\alpha}{({\tilde{I}})}=\lim_{s\rightarrow\infty}{}\frac{\alpha{({\tilde{I}}^{(s)})}}{s}=1+\frac{1}{n-2}=\frac{n-1}{n-2}}.$

∎

######  Remark 3.11.

In Corollaries 3.2 and 3.3, we proved the equality of ordinary and symbolic
powers for weighted oriented unicyclic graphs and complete graphs under
certain orientations and weights. If we change the orientation and weights, we
may lose the equality of ordinary and symbolic powers as it was shown in
Propositions 3.9 and 3.10.

$x_{1}$$x_{5}$$x_{2}$$x_{4}$$x_{3}$$2$$2$$2$$2$$2$$1$$2$$2$$x_{1}$$x_{5}$$x_{2}$$x_{4}$$x_{3}$$D_{1}$$D_{2}$
Figure 1. Two weighted oriented odd cycle’s of length $5$.

For example consider $D_{1}$ and $D_{2}$ be the weighted oriented odd cycles
as in Figure 1. Then
$I(D_{1})=(x_{1}x_{2}^{2},x_{2}x_{3}^{2},x_{3}x_{4}^{2},x_{4}x_{5}^{2},x_{5}x_{1}^{2})$
and
$I(D_{2})=(x_{1}x_{2}^{2},x_{2}^{2}x_{3},x_{3}x_{4}^{2},x_{4}^{2}x_{5},x_{1}x_{5}).$
In $D_{1},$ edges are naturally oriented and $w(x_{i})=2$ for each vertex
$x_{i}$. In $D_{2},$ vertices of $V^{+}(D_{2})=\\{x_{2},x_{4}\\}$ are sinks.
By Corollary 3.2, $I(D_{1})^{(s)}=I(D_{1})^{s}$ for all $s\geq 2$ where as by
Proposition 3.9, $I(D_{2})^{(s)}\neq I(D_{2})^{s}$ for all $s\geq 3$.

######  Proposition 3.12.

Let $\tilde{I}$ be the edge ideal of a weighted oriented graph $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is a clique sum of two odd cycles $C_{2n+1}=(x_{1},\ldots,x_{2n+1})$ and
$C_{2n+1}^{\prime}=(x_{1},y_{2},\ldots,y_{2n+1})$ of same length with a common
vertex $x_{1}$. Let $c=(\prod v|v\in V(C_{2n+1})\setminus
V^{+})(\prod{v^{w(v)}}|v\in V(C_{2n+1})\cap V^{+})$ and $c^{\prime}=(\prod
v|v\in V(C^{\prime}_{2n+1})\setminus V^{+})(\prod{v^{w(v)}}|v\in
V(C_{2n+1}^{\prime})\cap V^{+}).$

  1. (1)

Then ${\tilde{I}}^{(s)}={\tilde{I}}^{s}$ for $1\leq s\leq n$ and
${\tilde{I}}^{(n+1)}={\tilde{I}}^{n+1}+(c)+(c^{\prime}).$

  2. (2)

Let $s\in\mathbb{N}$ and write $s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and
$0\leq r\leq n.$ Then

${\tilde{I}}^{(s)}=\displaystyle{\sum_{p+q=t=0}^{k}{{\tilde{I}}^{s-t(n+1)}(c)^{p}(c^{\prime})^{q}}}.$

  3. (3)

If only one of the two cycles does not contain any element of $V^{+}$, then
for $s\in\mathbb{N}$, $\alpha{({\tilde{I}}^{(s)})}$
$\displaystyle{=2s-\Bigl{\lfloor}\dfrac{s}{n+1}\Bigr{\rfloor}}.$ In
particular, the Waldschmidt constant of ${\tilde{I}}$ is given by

$\displaystyle\widehat{\alpha}{({\tilde{I}})}=\frac{2n+1}{n+1}.$

  4. (4)

If both cycles contain some element of $V^{+}$, then for $s\in\mathbb{N}$,
$\alpha{({\tilde{I}}^{(s)})}$ $=2s.$ In particular, the Waldschmidt constant
of ${\tilde{I}}$ is given by $\widehat{\alpha}{({\tilde{I}})}=2.$

###### Proof.

Let $I$ be the edge ideal of $G$ and $\Phi$ is same as defined in Notation
3.6.

  1. (1)

Since $G$ is the clique sum of two odd cycles of same length with a common
vertex, by [4, Corollary 3.15], we have ${{I}^{(k)}}={{I}^{k}}$ for $1\leq
k\leq n$ and ${{I}^{(n+1)}}={{I}^{n+1}}+(x_{1}\cdots
x_{2n+1})+(x_{1}y_{2}\cdots y_{2n+1}).$ Thus by Theorem 3.7,
$\Phi({{I}^{(k)}})=\Phi({{I}^{k}}),$ i.e., ${\tilde{I}^{(k)}}={\tilde{I}^{k}}$
for $1\leq k\leq n$ and
$\Phi({{I}^{(n+1)}})=\Phi({{I}^{n+1}})+\Phi(x_{1}\cdots
x_{2n+1})+\Phi(x_{1}y_{2}\cdots y_{2n+1}),$ i.e.,
${\tilde{I}^{(n+1)}}={\tilde{I}^{n+1}}+(c)+(c^{\prime}).$

  2. (2)

Let $s\in\mathbb{N}$ and write $s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and
$0\leq r\leq n.$ By [4, Theorem 3.16], we have
$I^{(s)}=\displaystyle{\sum_{p+q=t=0}^{k}{I^{s-t(n+1)}(x_{1}\cdots
x_{2n+1})^{p}(x_{1}y_{2}\cdots y_{2n+1})^{q}}}.$ Thus by Theorem 3.7, applying
$\Phi$ on both sides we get
${\tilde{I}}^{(s)}=\displaystyle{\sum_{p+q=t=0}^{k}{{\tilde{I}}^{s-t(n+1)}(c)^{p}(c^{\prime})^{q}}}.$

  3. (3)

Assume only one of the both cycles does not contain any element of $V^{+}.$
Without loss of generality let $C_{2n+1}$ be the one without any element of
$V^{+}.$ Thus by (1),
${\tilde{I}^{(n+1)}}={\tilde{I}^{n+1}}+(c)+(c^{\prime})={\tilde{I}^{n+1}}+(x_{1}\cdots
x_{2n+1})+(c^{\prime}).$ Since all the edges of cycle $C_{2n+1}$ are without
any element of $V^{+},$ $\alpha(\tilde{I}^{s})=\alpha(I^{s})=2s$ for all $s.$
Here $\alpha(c)=2n+1$ and $\alpha(c^{\prime})\geq 2n+2.$ Thus we have
$\alpha(\tilde{I}^{(n+1)})=\alpha(I^{(n+1)})=2n+1.$ Since ${I}^{(s)}$ is
generated by ${I}$ and ${I}^{(n+1)},$ $\tilde{I}^{(s)}$ is generated by
$\tilde{I}$ and $\tilde{I}^{(n+1)}$ which implies
$\alpha{(\tilde{I}^{(s)})}=\alpha{({I}^{(s)})}.$ Therefore by [4, Theorem
3.12], we have
$\displaystyle{\alpha{(\tilde{I}^{(s)})}=2s-\Bigl{\lfloor}\dfrac{s}{n+1}\Bigr{\rfloor}}.$
Hence
$\displaystyle{\widehat{\alpha}{(\tilde{I})}=2-\frac{1}{n+1}=\frac{2n+1}{n+1}}.$

  4. (4)

Assume both cycles contain some element of $V^{+}.$ By (1), we have
${\tilde{I}}^{(s)}={\tilde{I}}^{s}+\displaystyle{\sum_{p+q=t=1}^{k}{{\tilde{I}}^{s-t(n+1)}(c)^{p}(c^{\prime})^{q}}}.$
Since $G$ contains odd cycles, there exists an edge in each odd cycle without
any element of $V^{+}.$ Thus $\alpha(\tilde{I}^{s})=\alpha(I^{s})=2s$ for all
$s.$ Since each of the both cycles contain some element of $V^{+},$
$\alpha(c)\geq 2(n+1),$ $\alpha(c^{\prime})\geq 2(n+1),$ $\alpha(c^{p})\geq
2p(n+1)$ and $\alpha({c^{\prime}}^{q})\geq 2q(n+1).$ Also
$\alpha(\tilde{I}^{s-t(n+1)})=2s-2t(n+1)$ and
$\alpha({\tilde{I}}^{s-t(n+1)}(c)^{p}(c^{\prime})^{q})\geq
2s-2t(n+1)+2p(n+1)+2q(n+1)=2s$ for $1\leq t\leq k.$ Since
$\alpha(\tilde{I}^{s})=2s,$
$\alpha(\tilde{I}^{(s)})=\alpha(\tilde{I}^{s})=2s.$ Hence
$\widehat{\alpha}{(\tilde{I})}=2.$

∎

##  4\. Regularity in weighted oriented odd cycle

In this section, we focus on weighted oriented odd cycles where the elements
of $V^{+}$ are sinks and we set $V^{+}\neq\phi.$ We first show that
$\operatorname{reg}\tilde{I}^{(s)}\leq\operatorname{reg}\tilde{I}^{s}$ and
then we explicitly compute
$\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})$. As a Corollary, we
conclude that the equality
$\operatorname{reg}\tilde{I}^{(s)}=\operatorname{reg}\tilde{I}^{s}$ occurs if
$V^{+}$ has only one element.

######  Lemma 4.1.

Let $\tilde{I}$ be the edge ideal of a weighted oriented cycle $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is the cycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $w_{i}=w(x_{i})$ where
$x_{i}\in V^{+},$ $w_{v}:=\max\\{w_{i}~{}|~{}x_{i}\in V^{+}\\}$ and
$\displaystyle{w=\sum_{x_{i}\in V^{+}}(w_{i}-1).}$ For $s\in\mathbb{N},$ let
$s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and $0\leq r\leq n.$ Then
$\mathfrak{m}^{(t-1)w_{v}+w}(c)^{t}\nsubseteq\tilde{I}^{t(n+1)}$ for $1\leq
t\leq k$ where $c=(\prod x_{j}|x_{j}\notin
V^{+})(\prod{x_{j}^{w_{j}}}|x_{j}\in V^{+})$ and
$\mathfrak{m}=(x_{1},\ldots,x_{2n+1})$ is the maximal homogeneous ideal.

###### Proof.

Here $c=(\prod x_{j}|x_{j}\notin V^{+})(\prod{x_{j}^{w_{j}}}|x_{j}\in V^{+})$
and $x_{v}$ be an element of $V^{+}$ with maximum weight. Here

$\tilde{I}=(x_{i}x_{j}^{w_{j}}~{}\mbox{if}~{}x_{j}\in
V^{+}~{}\mbox{or}~{}x_{i}x_{j}~{}\mbox{if}~{}x_{i},x_{j}\notin
V^{+}~{}|~{}\\{x_{i},x_{j}\\}\in E(C_{2n+1})).$

Let $\displaystyle{f=\prod_{x_{j}\in V^{+}}{x_{j}^{(w_{j}-1)}}}.$ Then
$\deg(f)=w$ and $f$ is a minimal generator of $\mathfrak{m}^{w}.$ Let
$g={x_{v}}^{(t-1)w_{v}}fc^{t}$ for some $t$ where $1\leq t\leq k.$ Then
$g\in\mathfrak{m}^{(t-1)w_{v}}\mathfrak{m}^{w}(c)^{t}=\mathfrak{m}^{(t-1)w_{v}+w}(c)^{t}.$
We want to prove that $g\notin\tilde{I}^{t(n+1)}.$

Note that $c$ can be expressed as the product of $n$ minimal generators of
$\tilde{I}$ and any $x_{i}$ where $x_{i}\notin V^{+}$ or the product of $n$
minimal generators of $\tilde{I}$ and any $x_{i}^{w_{i}}$ where $x_{i}\in
V^{+}.$ So the product of $c$ only with any $x_{j}^{w_{j}}$ for some $x_{j}\in
V^{+}$ or $x_{j}$ for some $x_{j}\notin V^{+}$ is exactly the product of $n+1$
minimal generators of $\tilde{I}.$ Hence the product of $c^{t}$ only with $t$
number of $x_{j}^{w_{j}}$ for some $x_{j}\in V^{+}$ or $x_{j}$ for some
$x_{j}\notin V^{+}$ is exactly the product of $t(n+1)$ minimal generators of
$\tilde{I}.$ Since $\displaystyle{f=\prod_{x_{j}\in
V^{+}}{x_{j}^{(w_{j}-1)}}}$ is neither a multiple of $x_{j}^{w_{j}}$ for some
$x_{j}\in V^{+}$ nor $x_{j}$ for some $x_{j}\notin V^{+},$
${x_{v}}^{(t-1)w_{v}}f$ can not be a multiple of product of $t$ number of
$x_{j}^{w_{j}}$ for some $x_{j}\in V^{+}$ or $x_{j}$ for some $x_{j}\notin
V^{+}$. Note that ${x_{v}}^{(t-1)w_{v}}f$ is a multiple of product of only
$t-1$ number of $x_{j}^{w_{j}}$ for $x_{j}=x_{v}\in V^{+}.$ Therefore
${x_{v}}^{(t-1)w_{v}}fc^{t}$ is not a multiple of product of any $t(n+1)$
minimal generators of $\tilde{I}.$ So
$g={x_{v}}^{(t-1)w_{v}}fc^{t}\notin\tilde{I}^{t(n+1)}.$ Hence
$\mathfrak{m}^{(t-1)w_{v}+w}(c)^{t}\nsubseteq\tilde{I}^{t(n+1)}$ for $1\leq
t\leq k$. ∎

######  Lemma 4.2.

Let $\tilde{I}$ be the edge ideal of a weighted oriented cycle $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is the cycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $w_{i}=w(x_{i})$ where
$x_{i}\in V^{+},$ $w_{v}:=\max\\{w_{i}~{}|~{}x_{i}\in V^{+}\\}$ and
$\displaystyle{w=\sum_{x_{i}\in V^{+}}(w_{i}-1).}$ For $s\in\mathbb{N},$ let
$s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and $0\leq r\leq n.$ Then
$\mathfrak{m}^{(k-1)w_{v}+w+1}\tilde{I}^{(s)}\subseteq\tilde{I}^{s}$ where
$\mathfrak{m}=(x_{1},\ldots,x_{2n+1})$ is the maximal homogeneous ideal.

###### Proof.

Let $c=(\prod x_{j}|x_{j}\notin V^{+})(\prod{x_{j}^{w_{j}}}|x_{j}\in V^{+}).$
Let $x_{v}$ be a vertex of $V^{+}$ with maximum weight. Here

$\tilde{I}=(x_{i}x_{j}^{w_{j}}~{}\mbox{if}~{}x_{j}\in
V^{+}~{}\mbox{or}~{}x_{i}x_{j}~{}\mbox{if}~{}x_{i},x_{j}\notin
V^{+}~{}|~{}\\{x_{i},x_{j}\\}\in E(C_{2n+1})).$

By Proposition 3.9, we have
$\tilde{I}^{(s)}=\tilde{I}^{s}+\displaystyle{\sum_{t=1}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}.}$
Since $\displaystyle{w+1=\sum_{x_{i}\in V^{+}}(w_{i}-1)}+1,$ any element of
$\mathfrak{m}^{w+1}$ can be viewed as a multiple of $x_{j}^{w_{j}}$ for some
$x_{j}\in V^{+}$ or a multiple of $x_{j}$ for some $x_{j}\notin V^{+}.$ Note
that the product of $c$ only with any $x_{j}^{w_{j}}$ for some $x_{j}\in
V^{+}$ or $x_{j}$ for some $x_{j}\notin V^{+}$ is exactly the product of $n+1$
minimal generators of $\tilde{I}.$ Thus any element of $\mathfrak{m}^{w+1}(c)$
can be represented as a multiple of product of $n+1$ minimal generators of
$\tilde{I}$ and hence $\mathfrak{m}^{w+1}(c)\subseteq\tilde{I}^{n+1}.$ First
we want to show that
$\mathfrak{m}^{w_{v}}\mathfrak{m}^{w+1}(c)\subseteq\mathfrak{m}^{w+1}\tilde{I}^{n+1}.$
Let $t_{1}\in\mathfrak{m}^{w_{v}}\mathfrak{m}^{w+1}(c)$ such that
$t_{1}=t_{2}t_{3}$ where $t_{2}\in\mathfrak{m}^{w_{v}}$ and
$t_{3}\in\mathfrak{m}^{w+1}(c).$ Since
$\mathfrak{m}^{w+1}(c)\subseteq\tilde{I}^{n+1},$ $t_{3}$ is a multiple of
product of $n+1$ minimal generators of $\tilde{I}.$ Let the product of those
$n+1$ minimal generators of $\tilde{I}$ is $t_{4},$ i.e.,
$t_{4}\in\tilde{I}^{n+1}.$ Then $t_{5}=t_{3}/t_{4}$ is a monomial of degree at
least $\displaystyle{\sum_{x_{i}\in V^{+}\setminus\\{x_{v}\\}}(w_{i}-1)}.$
Here $t_{2}t_{5}\in\mathfrak{m}^{w+1}$ because
$w_{v}+\displaystyle{\sum_{x_{i}\in V^{+}\setminus\\{x_{v}\\}}(w_{i}-1)}=w+1.$
So
$t_{1}=t_{2}t_{3}=t_{2}t_{4}t_{5}=(t_{2}t_{5})t_{4}\in\mathfrak{m}^{w+1}\tilde{I}^{n+1}.$
Thus
$\mathfrak{m}^{w_{v}}\mathfrak{m}^{w+1}(c)\subseteq\mathfrak{m}^{w+1}\tilde{I}^{n+1}.$
Hence

$\displaystyle\mathfrak{m}^{(t-1)w_{v}+w+1}(c)^{t}$
$\displaystyle=\mathfrak{m}^{(t-1)w_{v}}(c)^{t-1}\mathfrak{m}^{w+1}(c)$
$\displaystyle=\mathfrak{m}^{(t-2)w_{v}}(c)^{t-1}\mathfrak{m}^{w_{v}}\mathfrak{m}^{w+1}(c)$
$\displaystyle\subseteq\mathfrak{m}^{(t-2)w_{v}}(c)^{t-1}\mathfrak{m}^{w+1}\tilde{I}^{n+1}$
$\displaystyle=\mathfrak{m}^{(t-3)w_{v}}(c)^{t-2}\mathfrak{m}^{w_{v}}\mathfrak{m}^{w+1}(c)\tilde{I}^{n+1}$
$\displaystyle\subseteq\mathfrak{m}^{(t-3)w_{v}}(c)^{t-2}\mathfrak{m}^{w+1}\tilde{I}^{2(n+1)}$
$\displaystyle~{}~{}~{}~{}\vdots$
$\displaystyle\subseteq\mathfrak{m}^{w+1}(c)\tilde{I}^{(t-1)(n+1)}$
$\displaystyle\subseteq\tilde{I}^{n+1}\tilde{I}^{(t-1)(n+1)}$
$\displaystyle=\tilde{I}^{t(n+1)}.$

Therefore

$\displaystyle\mathfrak{m}^{(k-1)w_{v}+w+1}\tilde{I}^{s-t(n+1)}(c)^{t}$
$\displaystyle=\mathfrak{m}^{(k-t)w_{v}}\tilde{I}^{s-t(n+1)}\mathfrak{m}^{(t-1)w_{v}+w+1}(c)^{t}$
$\displaystyle\subseteq\mathfrak{m}^{(k-t)w_{v}}\tilde{I}^{s-t(n+1)}\tilde{I}^{t(n+1)}$
$\displaystyle\subseteq\mathfrak{m}^{(k-t)w_{v}}\tilde{I}^{s}$
$\displaystyle\subseteq\tilde{I}^{s}~{}\mbox{for}~{}1\leq t\leq k.$

Hence
$\mathfrak{m}^{(k-1)w_{v}+w+1}\tilde{I}^{(s)}=\mathfrak{m}^{(k-1)w_{v}+w+1}(\tilde{I}^{s}+\displaystyle{\sum_{t=1}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}})\subseteq\tilde{I}^{s}.$
∎

######  Theorem 4.3.

Let $\tilde{I}$ be the edge ideal of a weighted oriented cycle $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is the cycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $w_{i}=w(x_{i})$ where
$x_{i}\in V^{+},$ $w_{v}:=\max\\{w_{i}~{}|~{}x_{i}\in V^{+}\\}$ and
$\displaystyle{w=\sum_{x_{i}\in V^{+}}(w_{i}-1).}$ For $s\in\mathbb{N},$ let
$s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and $0\leq r\leq n.$ Then
$\operatorname{reg}\tilde{I}^{(s)}\leq\operatorname{reg}\tilde{I}^{s}.$

###### Proof.

Let $\mathfrak{m}=(x_{1},\ldots,x_{2n+1})$ is the maximal homogeneous ideal.
By Lemma 4.2, we have
$\mathfrak{m}^{(k-1)w_{v}+w+1}{\tilde{I}}^{(s)}\subseteq\tilde{I}^{s}.$ Thus
$\tilde{I}^{(s)}/\tilde{I}^{s}$ is an Artinian module. Therefore
$\dim(\tilde{I}^{(s)}/\tilde{I}^{s})=0$ and hence
$H_{\mathfrak{m}}^{i}(\tilde{I}^{(s)}/\tilde{I}^{s})=0$ for $i>0.$ Consider
the following short exact sequence

$0\rightarrow\tilde{I}^{(s)}/\tilde{I}^{s}\rightarrow
R/\tilde{I}^{s}\rightarrow R/\tilde{I}^{(s)}\rightarrow 0.$

Applying local cohomology functor we get
$H_{\mathfrak{m}}^{i}(R/\tilde{I}^{(s)})\cong
H_{\mathfrak{m}}^{i}(R/\tilde{I}^{s})$ for $i\geq 1$ and the following short
exact sequence

$0\rightarrow H_{\mathfrak{m}}^{0}(\tilde{I}^{(s)}/\tilde{I}^{s})\rightarrow
H_{\mathfrak{m}}^{0}(R/\tilde{I}^{s})\rightarrow
H_{\mathfrak{m}}^{0}(R/\tilde{I}^{(s)})\rightarrow 0.$ (1)

Now by the exact sequence $(\ref{exact}),$ we have
$a_{0}(R/\tilde{I}^{(s)})\leq a_{0}(R/\tilde{I}^{s}).$ Thus we can conclude
that
$\operatorname{reg}(R/\tilde{I}^{(s)})=\max\\{a_{i}(R/\tilde{I}^{(s)})+i~{}|~{}i\geq
0\\}\leq\max\\{a_{i}(R/\tilde{I}^{s})+i~{}|~{}i\geq
0\\}=\operatorname{reg}(R/\tilde{I}^{s}).$ Hence
$\operatorname{reg}\tilde{I}^{(s)}\leq\operatorname{reg}\tilde{I}^{s}.$ ∎

######  Theorem 4.4.

Let $\tilde{I}$ be the edge ideal of a weighted oriented cycle $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is the cycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $w_{i}=w(x_{i})$ where
$x_{i}\in V^{+},$ $w_{v}:=\max\\{w_{i}~{}|~{}x_{i}\in V^{+}\\}$ and
$\displaystyle{w=\sum_{x_{i}\in V^{+}}(w_{i}-1).}$ For $s\in\mathbb{N},$ let
$s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and $0\leq r\leq n.$ Then
$\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})=(s-n-1)(1+w_{v})+2w+2n+1$
for $s\geq n+1.$

###### Proof.

We fix $s\geq n+1.$ Let $c=(\prod x_{j}|x_{j}\notin
V^{+})(\prod{x_{j}^{w_{j}}}|x_{j}\in V^{+})$ and
$\mathfrak{m}=(x_{1},\ldots,x_{2n+1})$ be the maximal homogeneous ideal. Let
$x_{v}$ be an element of $V^{+}$ with maximum weight. By Lemma 4.2, we have
$\mathfrak{m}^{(k-1)w_{v}+w+1}\tilde{I}^{(s)}\subseteq\tilde{I}^{s}.$ Thus
$\tilde{I}^{(s)}/\tilde{I}^{s}$ is an Artinian module. Therefore
$\dim(\tilde{I}^{(s)}/\tilde{I}^{s})=0,$
$H_{\mathfrak{m}}^{0}(\tilde{I}^{(s)}/\tilde{I}^{s})=\tilde{I}^{(s)}/\tilde{I}^{s}$
and $H_{\mathfrak{m}}^{i}(\tilde{I}^{(s)}/\tilde{I}^{s})=0$ for $i>0.$ So
$\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})=a_{0}(\tilde{I}^{(s)}/\tilde{I}^{s}).$
We want to prove that one of the maximum degree element of $I^{(s)}$ which
does not lie in $\tilde{I}^{s}$ will come from the ideal
$\tilde{I}^{s-(n+1)}(c)^{1}$.
By Proposition 3.9, we have
$\tilde{I}^{(s)}=\tilde{I}^{s}+\displaystyle{\sum_{t=1}^{k}\tilde{I}^{s-t(n+1)}(c)^{t}.}$
Here $\deg(c)=2n+1+w$ and $\deg(c^{t})=t(2n+1+w).$ Without loss of generality
we can assume that $x_{v}\neq x_{2n+1}$. Then $x_{v}^{w_{v}}x_{v+1}$ is a
maximum degree minimal generator of $\tilde{I}$ which implies that one maximum
degree minimal generator of $\tilde{I}^{s-t(n+1)}$ is
$(x_{v}^{w_{v}}x_{v+1})^{s-t(n+1)}$ and its degree is $(s-t(n+1))(w_{v}+1).$
From the proof of Lemma 4.2, we observe that
$\mathfrak{m}^{(t-1)w_{v}+w+1}(c)^{t}\subseteq\tilde{I}^{t(n+1)}$ for $1\leq
t\leq k.$ Thus by Lemma 4.1, the maximum value of $u$ such that
$\mathfrak{m}^{u}(c)^{t}\nsubseteq\tilde{I}^{t(n+1)}$ is $(t-1)w_{v}+w$ for
$1\leq t\leq k.$ So the maximum degree of an element of the ideal $(c)^{t}$
which does not lie in $\tilde{I}^{t(n+1)}$ is $(t-1)w_{v}+w+t(2n+1+w)$ for
$1\leq t\leq k.$ Hence the maximum degree of an element of
$\tilde{I}^{s-t(n+1)}(c)^{t}$ which does not lie in
$\tilde{I}^{s-t(n+1)}\tilde{I}^{t(n+1)}=\tilde{I}^{s}$ is
$(s-t(n+1))(w_{v}+1)+(t-1)w_{v}+w+t(2n+1+w)$ for $1\leq t\leq k.$ Let
$d(t)=(s-t(n+1))(w_{v}+1)+(t-1)w_{v}+w+t(2n+1+w)$ for $1\leq t\leq k.$ If
$t_{1}<t_{2}$, then

$\displaystyle d(t_{1})-d(t_{2})$
$\displaystyle=(s-t_{1}(n+1))(w_{v}+1)+(t_{1}-1)w_{v}+w+t_{1}(2n+1+w)$
$\displaystyle\hskip
8.5359pt-(s-t_{2}(n+1))(w_{v}+1)-(t_{2}-1)w_{v}-w-t_{2}(2n+1+w)$
$\displaystyle=(t_{2}-t_{1})(n+1)(w_{v}+1)-(t_{2}-t_{1})w_{v}-(t_{2}-t_{1})(2n+1+w)$
$\displaystyle=(t_{2}-t_{1})((n+1)(w_{v}+1)-w_{v}-(2n+1+w))$
$\displaystyle=(t_{2}-t_{1})(n(w_{v}+1)-(2n+w))$
$\displaystyle=(t_{2}-t_{1})(n(w_{v}-1)+2n-2n-w)$
$\displaystyle=(t_{2}-t_{1})(n(w_{v}-1)-w).$

Since $|E(D)|=2n+1,$ there are at most $n$ elements of $V^{+}.$ So
$n(w_{v}-1)\geq\displaystyle{\sum_{x_{i}\in V^{+}}(w_{i}-1)}=w$ which implies
that $d(t_{1})-d(t_{2})\geq 0.$ Thus $d(1)\geq d(2)\geq\cdots\geq d(k).$
Therefore one of the maximum degree elements of $\tilde{I}^{(s)}$ which does
not lie in $\tilde{I}^{s}$ will come from $\tilde{I}^{s-(n+1)}(c)^{1}$ for
$t=1$ and its degree is
$d(1)=(s-n-1)(w_{v}+1)+w+(2n+1+w)=(s-n-1)(w_{v}+1)+2w+2n+1.$ Hence
$\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})=a_{0}(\tilde{I}^{(s)}/\tilde{I}^{s})=(s-n-1)(w_{v}+1)+2w+2n+1$
for $s\geq n+1.$ ∎

######  Corollary 4.5.

Let $\tilde{I}$ be the edge ideal of a weighted oriented cycle $D$ where the
elements of $V^{+}$ are sinks and let $G$ be the underlying graph of $D$ which
is the cycle $C_{2n+1}=(x_{1},\ldots,x_{2n+1}).$ Let $V^{+}=\\{x_{v}\\}$ and
$w_{v}=w(x_{v})$ for some $x_{v}\in V(C_{2n+1}).$ For $s\in\mathbb{N},$ let
$s=k(n+1)+r$ for some $k\in\mathbb{Z}$ and $0\leq r\leq n.$ Then
$\operatorname{reg}\tilde{I}^{(s)}=\operatorname{reg}\tilde{I}^{s}$ for all
$s.$

###### Proof.

Let $I$ be the edge ideal of $G$ and $\Phi$ is same as defined in Notation
3.6. First we fix $s\geq n+1.$ Consider the following short exact sequence

$0\rightarrow\tilde{I}^{(s)}/\tilde{I}^{s}\rightarrow
R/\tilde{I}^{s}\rightarrow R/\tilde{I}^{(s)}\rightarrow 0.$ (2)

Without loss of generality we can assume that $V^{+}=\\{x_{1}\\}.$ Thus by
Theorem 4.4, we have
$\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})=(s-n-1)(w_{1}+1)+2(w_{1}-1)+2n+1.$
Here $x_{2}x_{1}\in I$ and by definition of $I^{(s)},$ we have
$(x_{2}x_{1})^{s}\in{I}^{(s)}.$ Observe that $(x_{2}x_{1})^{s}$ is a minimal
generator of ${I}^{(s)}.$ Then by Theorem 3.7,
$\Phi((x_{2}x_{1})^{s})=(x_{2}x_{1}^{w_{1}})^{s}\in\tilde{I}^{(s)}$ and
$(x_{2}x_{1}^{w_{1}})^{s}$ is a minimal generator of degree $s(w_{1}+1)$ in
$\tilde{I}^{(s)}.$ Since there exist a minimal generator of degree
$s(w_{1}+1)$ in $\tilde{I}^{(s)},$ we have

$\displaystyle\operatorname{reg}(\tilde{I}^{(s)})$ $\displaystyle\geq
s(w_{1}+1)$ $\displaystyle=(s-n-1)(w_{1}+1)+(n+1)(w_{1}-1+2)$
$\displaystyle=(s-n-1)(w_{1}+1)+(n+1)(w_{1}-1)+2(n+1)$
$\displaystyle>\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s})$

as $n\geq 1.$ So
$\operatorname{reg}(R/\tilde{I}^{(s)})\geq\operatorname{reg}(\tilde{I}^{(s)}/\tilde{I}^{s}).$
Thus by the exact sequence (2) and [12, Lemma 1.2], we have
$\operatorname{reg}(R/\tilde{I}^{s})\leq\operatorname{reg}(R/\tilde{I}^{(s)}).$
Also by Theorem 4.3, we have
$\operatorname{reg}(R/\tilde{I}^{(s)})\leq\operatorname{reg}(R/\tilde{I}^{s}).$
Therefore
$\operatorname{reg}(R/\tilde{I}^{(s)})=\operatorname{reg}(R/\tilde{I}^{s}),$
i.e., $\operatorname{reg}\tilde{I}^{(s)}=\operatorname{reg}\tilde{I}^{s}$ for
$s\geq n+1.$ By Proposition 3.9, we have
$\operatorname{reg}\tilde{I}^{(s)}=\operatorname{reg}\tilde{I}^{s}$ for $1\leq
s\leq n.$ Hence
$\operatorname{reg}\tilde{I}^{(s)}=\operatorname{reg}\tilde{I}^{s}$ for all
$s.$

∎

Acknowledgements
The first author was supported by SERB (grant No.: EMR/2016/006997), India.
The authors are grateful to the referee for his helpful comments and
suggestions.

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