# Modular Design Patterns for Hybrid Actors

André Meyer-Vitali, Wico Mulder, Maaike H.T. de Boer
TNO, Netherlands Organisation for Applied Scientific Research
Anna van Buerenplein 1, 2595 DA, The Hague, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Recently, a boxology (graphical language) with design patterns for hybrid AI
was proposed, combining symbolic and sub-symbolic learning and reasoning. In
this paper, we extend this boxology with actors and their interactions. The
main contributions of this paper are: 1) an extension of the taxonomy to
describe distributed hybrid AI systems with actors and interactions; and 2)
showing examples using a few design patterns relevant in multi-agent systems
and human-agent interaction.

##  1 Introduction

A boxology is a graphical language using boxes and (directed) lines, such as
used in many types of diagrams. In earlier work, we presented a boxology [3]
and stressed the use of design patterns to combine different paradigms in AI:
knowledge-based reasoning and data-driven machine learning. In this paper we
extend on this and include the aspect of actors, along with the corresponding
interaction with processes, instances and models.

Design patterns can be visualised as a boxology, which is a representation of
an organized structure in the form of a directed graph of labeled nodes,
called boxes, and their relations and dependencies as arrows. Using a boxology
improves communication and understanding of the structure of AI systems. This
work aims to make complex distributed AI systems more transparent in terms of
their design patterns. The modularity of the proposed design patterns also
facilitates the design and engineering of complex AI systems. This allows for
including descriptions of interactions among autonomous entities, such as
software agents or robots. As a result, the boxology can be of relevance to a
broader spectrum of distributed AI applications and topics such as multi-agent
systems, federated or multi-agent learning, human-agent teaming and social
intelligence.

An emerging field in AI is the combination of data-driven and knowledge-driven
approaches [1]. In addition, sharing of knowledge (collaborative learning and
reasoning) and the collaboration between humans and actors remains a paramount
challenge for truly hybrid AI. Extending the boxology with actors allows for
specifying the aspects of human-machine interactions in such systems properly.
We expect that the boxology approach may support discussions on autonomy,
explainability, trustworthiness and sovereignty of AI systems. Furthermore,
the inclusion of actors may help to address the responsibility and ownership
of models and processes; topics that are being encountered on top of technical
engineering questions - and of increasing importance in complex distributed AI
systems. To the best of our knowledge no conceptual framework is available in
which design elements can be discussed, compared, configured and combined with
respect to those topics.

A framework describing a set of modular design patterns was initially
developed by van Harmelen and ten Teije [2], and extended by van Bekkum et al.
[3]. In this paper, we use that conceptual framework as a basis (section 2)
and extend it with the missing pieces, namely actors and interactions (section
3). In section 4, we show examples of using the design patterns with actors.
We finish the paper with conclusions and directions for future work (section
5).

###  1.1 Agency

Artificial Intelligence is based on the principles of autonomy and agency.
Without them, intelligence cannot manifest itself. Agency is the capacity of
individuals to act independently and to make their own free choices, according
to Kant and others [6]. As mentioned above, (Artificial) Intelligence is
unthinkable without agency. First of all, a minimal degree of autonomy is
required to avoid purely predictable and reactive behaviour. Intelligence is,
in fact, the result of a lot of interaction - be it in physiological (brains)
or social systems (swarms) - and forms the basis and inspiration of most AI
techniques (neural networks, knowledge graphs, communication networks). In the
"Society of Mind" [24] Minsky gave a number of examples for distributed AI and
the importance of the interactive/social aspects of intelligence. Intelligence
as a social and sociological phenomenon implies the formulation of intentional
behaviour [25], where intents guide the planning of (inter-)actions. However,
the control of social systems is not centralised, but shared among all
participants, which leads to emergent behaviour of the system as a whole. As a
result, individual autonomous actors need to consider and reflect upon the
effect of their actions on other actors and their mental states, such as their
knowledge, intentions, and beliefs (theory of mind [26] [27]), as well as
their virtual or physical context (contextual embedding).

##  2 Taxonomy & Patterns

In order to describe design patterns of AI systems, a common terminology is
required that defines a hierarchical taxonomy of processes and their inputs
and outputs on various levels of abstraction. On the highest level of
abstraction the concepts of Instance, Model, Process and Actor are defined
(see figure 1). The full taxonomy is larger and more detailed. For example,
data and symbols, as well as induction and deduction, have a few subconcepts
that are not shown here for the sake of clarity and space.

Instances are the basic building blocks of “things”, examples or single
occurrences of something. The two main classes of instances are data (numbers,
texts, tensors, streams) and symbols (labels, relations, traces). In contrast
to data, symbols must designate objects, classes or relations in the world.
There must be compositional rules and a system of operations to generate new
symbols [4].

Models are descriptions of entities and their relationships. They are useful
for inferring data and knowledge. Models in hybrid AI systems can be either
(a) statistical models, such as (Deep) Neural Networks, Bayesian Networks and
Markov Models, or (b) semantic models, such as Taxonomies, Ontologies and
Knowledge Graphs.

In order to perform operations on instances and models, processes define the
steps that lead from inputs to results. The main types of processes are: (a)
the transformation of instances and models, e.g. embedding in a vector space,
and (b) inferencing, which includes induction or deduction. Induction is the
generation of models either by means of generalising data using statistical
machine learning algorithms or by means of knowledge engineering with domain
experts. Deduction is the use or application of models and related instances
for arriving at classifications, predictions and other kinds of conclusions,
such as reasoning.

Figure 1: Taxonomy for Hybrid AI (simplified)

In section 3 of this paper we work out the main concept Actor and the related
subconcepts added to processes (Interaction), a few semantic models (State,
Norm and Context) and symbols (Request, Reply, etc.).

With the basic taxonomical terms in place, design patterns can be defined. The
patterns can be used to describe different levels of abstraction: elementary
patterns can be used in design descriptions, and more detailed patterns
support architectural choices on types of models and instances used in an AI
system.

Figure 2: Machine Learning Pipeline

The main concepts and their subconcepts are visualised in diagrams using
rectangles for instances, hexagons for models, ovals for processes and
triangles for actors. The main category (as a verb) and the lowest subcategory
in the hierarchy are used within the boxes separated with a colon, such as
infer:deduce (where deduction is a subconcept of inference). Figure 2
describes the typical Machine Learning pipeline in terms of the taxonomy. This
more complex pipeline (pattern 3a) consists of 2 elementary patterns (1a) and
(2a) (not shown). Pattern (1a, top row in figure 2) describes the standard
training process, in which data is used to train a model (model generation).
Pattern (2a) describes the application process, where (new) data is fed to the
model and inferencing is performed to output a symbol, such as a
classification. Hence, "Machine Learning" involves both an inductive step
(generation by training) to produce a statistical model and a deductive step
to use that model for classification or prediction.

##  3 Extending the Boxology with Actors

The concept of actors was included in the previous version of the boxology.
However, it was not yet worked out in detail, because the focus was on
combining knowledge-driven reasoning with data-driven learning.

Actors in a hybrid AI system are autonomous entities that can be either
humans, agents [7] [8] [11] or teams. Agents can take several forms, such as
software agents (e.g., apps, services, mobile agents) or robots (physically
embedded agents, e.g., drones or (parts of) autonomous vehicles). In the
manufacturing domain actors can be cyber-physical systems, i.e., systems that
consist of a physical part (e.g., a machine on the shopfloor) and a software
part, often referred to as a digital twin [9] [10]. In fact, there is renewed
interest in distributed and multi-agent systems, be it in robotics or for the
semantic web [5], because they fit perfectly in current system architectures
and infrastructures. Peer-to-peer networks (p2p), cloud services, edge
computing, Internet of Things (IoT) and federated or multi-agent reasoning and
learning (MARL) are perfect use cases for agent technology. They all make use
of distributed systems with decentralised data and processing. In all these
systems, coordination is necessary - either collaboratively or competitively -
and agent technology has many proven solutions to these challenges. The actor
concept, according to our taxonomy, includes robots (such as automated
vehicles), software and human agents, as well as hybrid teams of all of them.

###  3.1 Autonomous Actors

The relation between an actor and a process often comes with the notion of its
role or responsibility. An actor can initiate a process or support an existing
process. Sometimes roles typically characterise an actor, e.g., when actors
simply fulfill a particular designated task such as a robot, a data-provider
in a data ecosystem, or a machine on a manufacturing shopfloor. Actors may
also fulfill multiple roles to either the same or to different processes.
Examples of such actors are digital twins or human machine operators.

Actors may interact with each other and form teams in order to coordinate
tasks or learn to reach a common goal [15]. Examples are swarms, multi-agent
systems and human-agent teams. A team is a group of collaborating entities
that can itself be regarded as a single actor [12]. The aggregated team actors
simplify the interaction models for many-to-many coordination by representing
common goals and shared knowledge and resources (see below for examples).

Actors interact with each other using diverse (internal) behaviours and
coordination protocols (subconcepts of Interaction), such as collaboration,
cooperation or competition [13]. The interactions between actors define the
so-called collective intelligence and may lead to emergent behaviour. As
explained and demonstrated in the patterns below, team actors aggregate
multiple concurrent lines of interaction concisely.

Autonomy is a gradual property [14], ranging from remotely controlled to
selfish behaviour and all forms of pro-active behaviour in-between. It is
subject to commitments made among agents: the more commitments an agent makes,
the more it voluntarily limits its own autonomy temporarily. The rational and
intentional scalability of agents’ autonomous behaviour allows for various
forms of coordination and team work. Intentionally and temporarily reducing
its autonomy may well be in the best interest of the agent to achieve goals
that cannot be achieved alone. Alternatively, a team can achieve more than
each individual actor when actors consciously contribute to achieving the
common team goal. This is when various forms of team coordination become
essential.

##  4 Applications

In order to illustrate the above-mentioned concepts, three exemplary design
patterns and a use case are explained below.

It is important to mention that the boxology allows to represent both the
interactions among actors, as well as the internal structural patterns of AI
systems. Consequently, it supports the design of hybrid AI systems using many
different interaction, learning and reasoning patterns.

In order to visualise both aspects of interactive systems, a new design
component was required for the diagrams. We chose for including a zooming
function that is shown as a rectangular frame with a solid line and a light
grey background (not to be confused with the patterns as shown with a dashed
frame and dark grey background, see figure 2). A badge at the top left of the
frame indicates the type and instance of the component that is detailed inside
the frame. Here, a triangular badge indicates that an actor is zoomed in. The
same mechanism can also be used for zooming into processes, models or
instances. In the case that an actor represents a team, it is possible to
include individual actors within the frame as members of that team. This
nesting is not suitable for individual actors, of course, but works for sub-
processes, sub-models and sub-instances.

###  4.1 Multi-Agent Learning and Reasoning

Interactive actors can collaborate for learning and reasoning and to share
knowledge [16]. In figure 3 a multi-agent approach is shown where one actor
requests a team to learn partial models using a provided learning algorithm,
rather than collecting all data and processing it centrally. The learning
algorithm is embedded in the requests (as a code model) and the partial model
is returned in the responses. This approach is more efficient by allowing for
parallel processing and it avoids privacy restrictions. In alternative
approaches (not shown), an actor can simply send a request message and let the
team members learn according to their own methods. Yet another approach is to
use mobile agents that migrate to a target platform, process the data locally
and move on with a trained model to other platforms. In all these patterns,
partial models (models Ti) are integrated at some point by one designated
party.

A related scenario is learning on demand, where an actor asks another actor
for knowledge that the second one may already have learned and the requester
needs at a certain point in time. For example, the second actor can teach the
first one to perform a certain task.

Figure 3 shows two parties: Actor A1 and Team A. This is a very compact and
concise representation of group behaviour. However, there can be more roles,
e.g. a coordinator, that would be represented explicitly. Therefore, it is
possible to represent more complex use cases where multiple roles can be used
- eventually as subteams if necessary. In the case, as shown, each actor
behaves according to the same interactive principles, thereby demonstrating a
distributed interaction pattern without any central control (peer-to-peer).

Figure 3: Multi-Agent Learning with Mobile Code Figure 4: Distributed Planning
using Job, Pool and Machine Agents

###  4.2 Use Case: Multi-Agent Planning

In a manufacturing use case we use a similar approach for distributed
planning. The boxology extends other engineering diagrams and focuses
specifically on the underlying machine learning parts of the system. This is
shown in figure 4.

The distributed system consists of three types of software agents: job agents,
pool agents and machine agents. The architecture consists of 4 parts. Part I
shows that a Job Agent uses a bill of material model that was generated by a
human operator. It takes product orders from an Enterprise Resource Planning
(ERP) system and creates work orders. The work orders are sent to a pool
agent, which controls a set of production machines. This is shown in part II.
Note that the attention in the diagram is on the job agent, therefore the
actor representing the job agent is placed in the corner of the grey box. The
role of the operator is merely supportive here, as it generates a model of the
factory shop floor. The pool agent receives requests for work orders. Using
the model that was generated by a human operator, the job agent turns a work
order into a specific job that can be carried out by one or more machines on
the shop floor. Every eligible machine agent in the pool receives that job
request and uses its own capacity model to judge whether it can perform the
job in the requested time. If so, it informs the other machines in the pool
about a possible time slot. This is shown in part III. The interaction between
the machines is a bidding process. The machine with the best option, i.e.,
that has offered highest bid, takes the job, updates its capacity model and
informs the job agent. Part IV of the diagram shows the role of the team in
that bidding process. The exact steps of this bidding process are not shown
here. For the example, it suffices to show the main interaction between a
machine agent and its team.

Eventually an incoming time slot leads via a bidding mechanism to the
assignment of the job. The machine agent that will carry out the job will be
informed, and the team (all the machine agents) updates the capacity model.
Note, that part IV shows a composite actor (the team) in its interaction with
a particular agent. The boxology is used to abstract from details such as the
bidding. A process which often deserves its own details either in a boxology
diagram (or a sub-diagram) or in a different type of interaction diagram as
preferred by the system architect.

###  4.3 Contextual Actors

Situated or contextual actors are interacting with their physical or virtual
environment. Various models exist for specifying the necessary steps and
interactions for context-aware actors, such as BDI (beliefs, desires,
intentions) [17] or OODA (observe, orient, decide, act) [18]. For the
interaction with their environment, actors perform communicative acts, namely
sensing and acting. Actors can also perform communicative acts with other
actors using language (speech acts, see section 4.4).

For contextual or situated actors, the concept of Deduction is extended with
Planning and Interaction with communicative acts [19] [20]: Sensation, Action
and Speech. These subconcepts will be used in the patterns below. For certain
interactions, additional symbols are required, such as Request, Proposal,
Assignment, or Result. They play roles in interaction protocols, as defined by
FIPA (see section 4.4 below), and other communicative acts.

Relevant model-related concepts for actors are the semantic models of State,
Context and Norm [21]. State includes reasoning components, such as beliefs,
desires and intentions (to be used in BDI reasoning, see section 4.3 below).
Context represents the external world (physical and virtual) that an actor can
sense and act upon. An important aspect of the context (or environment) are
resources. A Resource model specifies the availability and capacity of objects
and actors. For example, time limits or cognitive load could be relevant.
Norms define and regulate interaction among actors according to desirable and
permitted behaviour (physical, legal, social or otherwise).

Figure 5 shows a contextually reasoning BDI actor. The actor senses data from
the environment, classifies data to symbolic beliefs, predicts desires, plans
intentions and acts on the environment. These processes make use of dedicated
models, namely a world model (beliefs), a model of goals (desires) and a model
of plans (intentions).

Besides acting on the environment, the actions can also take the form of
speech acts to communicate with other actors of a team. This is indicated
(without going into details) using the dotted arrows in the figure to show the
incorporation of team knowledge that is shared via speech acts. Consequently,
actors of a team can influence other actors’ world and goal models (not
without consent, of course).

Figure 5: Contextual Agent using BDI Figure 6: Team Negotiation with
ContractNet

###  4.4 Negotiation

A well-known class of interactions are negotiations. As an example, figure 6
shows a negotiation based on ContractNet [22], as specified by FIPA [23]. The
planning steps of each agent (and team member) can be specified in more detail
with more specific (induction and) deduction methods and corresponding models.
Depending on the type of model learned in an inductive step, the deductive
planning steps can use different strategies to come to concrete replies in a
given situation.

The negotiation follows the above-mentioned interaction protocol and uses
speech acts for the individual steps of the protocol. A call for proposals
(CFP) is sent out by an actor to a team that it belongs to. Each team member
may make a proposal (within a certain time limit) and the originator can
accepts and reject them. When receiving an acceptance notification with an
assignment, actors will produce a compliant result and return this to the
requesting actor.

The pattern in figure 6 does not specify the exact means of communication
among the actors. Interaction, containing a CFP, for example, could occur via
email, RMI, or any other message passing interface and infrastructure.
However, the level of detail and abstraction can vary according to the
maturity of the design. The goal is to define design patterns at a relatively
high level of abstraction and to use a few of these for composing a hybrid AI
system. Subsequently, the concepts can be made increasingly concrete. For
example, choices will have to be made for the exact models and means of
communication. From there, deployment of the system can be automated using off
the shelf components.

##  5 Conclusion and Future Work

In this paper, we extend the previous hybrid AI boxology with actors and
interactions to make it usable in the field of distributed hybrid AI and
human-agent interaction. We show three design patterns relevant in this field,
where interaction among various actors is given shape and where actors learn
and reason about their interactions. We also show a use case where agents
interact in planning a production task. The visualisation of internal and
external processes facilitates the identification and reuse of design
patterns.

In future work, we plan to extend the taxonomy and the patterns even further,
and to use the patterns for actually designing and efficiently implementing
distributed hybrid AI systems. By avoiding a black box approach and explicitly
describing the components and their interactions, such systems are expected to
become more transparent, explainable and verifiable (predictable and
reproducible). The extended boxology allows designers of distributed hybrid AI
to produce more trustworthy systems.

## Acknowledgement

This research was partially supported by TAILOR, a project funded by the EU
Horizon 2020 research and innovation programme under GA No 952215.

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# Regularity of power of edge ideal of Cohen-Macaulay weighted oriented
forests

Manohar Kumar∗ Department of Mathematics, Indian Institute of Technology
Kharagpur, West Bengal, INDIA - 721302<EMAIL_ADDRESS>and  Ramakrishna
Nanduri† Department of Mathematics, Indian Institute of Technology Kharagpur,
West Bengal, INDIA - 721302<EMAIL_ADDRESS>
###### Abstract.

In this paper, we explicitly give combinatorial formulas for the regularity of
powers of edge ideals, $\operatorname{reg}(I(D)^{k})$, of weighted oriented
unmixed forests $D$ whose leaves are sinks ($V^{+}(D)$ are sinks). This
combinatorial formula is a piecewise linear function of $k$, for $k\geq 1$.

∗ Supported by PMRF fellowship, India

† Supported by SERB grant No: CRG/2021/000465, India

AMS Classification 2010: 13D02, 05E99, 13D45, 13A30, 05E40

##  1\. Introduction

A weighted oriented graph or a digraph is a graph $D=(V(D),E(D),w)$, where
$V(D)$ is the vertex set of $D$, $E(D)=\\{(x,y)|\mbox{ there is an edge from
vertex $x$ to vertex $y$}\\}$ is the edge set of $D$, and
$w:V(D)\rightarrow\mathbb{N}$ is a map, called weight function. That is,
assign a weight $w(x)$ to each vertex $x$ of $D$. If $D$ is a weighted
oriented graph, then the underlying simple graph $G$ whose $V(G)=V(D)$ and
$E(G)=\\{\\{x,y\\}|(x,y)\in E(D)\\}$. That is, $G$ is the simple graph without
orientation and weights in $D$. Let $V(D)=\\{x_{1},\ldots,x_{n}\\}$ and
$R=\mathbb{K}[x_{1},\ldots,x_{n}]$, a polynomial ring in $n$ variable
$x_{1},\ldots,x_{n}$, where $\mathbb{K}$ is a field. Then the edge ideal of
$D$ is defined as the ideal

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

Studying the edge ideals of weighted oriented graphs is an active area of
research at present times. Currently, many authors are studying various
algebraic invariants of $I(D)$, namely, Castelnuovo-Mumford regularity, depth,
projective dimension, etc. Weighted oriented graphs are important because of
their applications in coding theory; see [10, 17]. The connection between the
algebraic invariants and the combinatorial invariants of a weighted oriented
graph is an important problem in combinatorial Commutative Algebra. In
particular, describing an algebraic invariant associated to $I(D)$ in terms of
certain combinatorial information of $D$ is a challenging problem. It is well
known that $\operatorname{reg}(I(D)^{k})$ is eventually a linear function of
$k$, see [3, 15]. It is very hard to find this linear function and very few is
known in the literature.

Gimenez, Martínez-Bernal, Simis and Villarreal gave a combinatorial
characterization of Cohen-Macaulay weighted oriented forests, see [7, Theorem
5]. In [17], Pitones, Reyes and Toledo characterized associated primes and the
unmixed property of $I(D)$. Also, see [5] for a characterization of
unmixedness of certain weighted oriented graphs. The projective dimension and
regularity of $I(D)$ were explicitly computed by Zhu, Xu, Wang, and Tang in
[20], where $D$ is a weighted rooted forest or an oriented cycle by assuming
all the weights of the vertices are at least $2$. The invariants of
$I(D)^{k}$, where $D$ is a weighted oriented gap-free bipartite graph, have
explicit combinatorial formulas provided by Zhu, Xu, Wang, and Zhang in [21].
Also see [18]. For weighted oriented rooted forests $D$, the regularity and
projective dimension of $I(D)^{k}$ are computed in [19] by assuming all the
weights of the vertices are at least $2$. For $m$-partite weighted oriented
graphs, formulas for those invariants are explicitly given in [22]. In [4],
Betti numbers of $I(D)$ are explicitly computed for a class of weighted
oriented graphs $D$. Also, see [16] to refer formulas for regularity of a
class of weighted oriented graphs whose vertices have weights at least $2$.
For a weighted oriented path or a cycle whose edges are oriented in one
direction, then its regularity was computed in [14]. In this work, we proved
that if $D$ is a weighted oriented Cohen-Macaulay forest with all its leaves
are sinks, then we explicitly compute combinatorial formulas for regularity of
$I(D)^{k}$ for all $k\geq 1$ and no assumptions on the weights of the
vertices. This formula for $\operatorname{reg}(I(D)^{k})$ is eventually a
linear function of $k$. Our approach does not have any assumptions on the
weights.

Now we give a section-wise description of the paper. In section 2, we recall
the definitions and the basic results that we will use in the sequel. In
section 3, we compute $\operatorname{reg}(I(D))$, where $D$ is a weighted
oriented Cohen-Macaulay forest with all its leaves are sinks (Theorem 3.6) and
we prove some technical lemmas. Finally in section 4, we prove a combinatorial
formula for $\operatorname{reg}(I(D)^{k})$ for any $k\geq 1$ (Theorem 4.3).

##  2\. Preliminaries

In this section, we recall definitions and some results which will be used
throughout the paper. Also, we set up notation.
Let $D=(V(D),E(D),w)$ be a weighted oriented graph and
$V(D)=\\{x_{1},\ldots,x_{n}\\}$. Let $R=\mathbb{K}[x_{1},\ldots,x_{n}]$, where
$\mathbb{K}$ is a field. Let $V^{+}(D):=\\{x\in V(D):w(x)\geq 2\\}$ and simple
denoted by $V^{+}$. If $e=(x,y)\in E(D)$, then $x$ is called head of $e$ and
$y$ is called tail of $e$. For a vertex $x\in V(D)$, its outer neighbourhood
is defined as $N_{D}(x)^{+}:=\\{y\in V(D)|(x,y)\in E(D)\\}$ and its inner
neighbourhood is defined as $N_{D}(x)^{-}:=\\{z\in V(D)|(z,x)\in E(D)\\}$ and
$N_{D}[x]:=N_{D}(x)^{+}\cup N_{D}(x)^{-}\cup\\{x\\}$,
$N_{D}[x]^{+}:=N_{D}(x)^{+}\cup\\{x\\}$,
$N_{D}[x]^{-}:=N_{D}(x)^{-}\cup\\{x\\}$. A vertex $x\in V(D)$ is called a
source if $N_{D}(x)^{-}=\emptyset$ and $x$ is called a sink if
$N_{D}(x)^{+}=\emptyset$. If $x\in V(D)$ is a source, then we set $w(x)=1$.
The degree of a vertex $x\in V(D)$, is defined as $d_{D}(x):=|N_{D}(x)|$. For
any $S\subseteq V(D)$, we denote $D\setminus S$, the induced subgraph on the
vertex set $V(D)\setminus S$. For a monomial $f\in R$, define its support as
$\operatorname{supp}(f):=\\{x_{i}:x_{i}\mid f\\}$. We denote $\mathcal{G}(I)=$
minimal set of generators of $I$. For a monomial ideal $I$, its support is
defined as
$\operatorname{supp}(I):=\displaystyle\cup_{f\in\mathcal{G}(I)}\operatorname{supp}(f)$.

Recall that two edges $\\{x,y\\},\\{z,u\\}\in E(G)$ of a simple graph $G$ are
said to be adjacent if there exists an edge between them, that is there exists
an edge $\\{a,b\\}\in E(G)$, where $a\in\\{x,y\\}$ and $b\in\\{z,u\\}$. Two
edges in a weighted oriented graph are said to be adjacent if they are
adjacent in its underlying graph. We denote $[r]=\\{1,\ldots,r\\}$, for any
positive integer $r$. Recall that a weighted oriented graph is said to be a
forest (or tree) if its underlying graph is a forest(or tree). We say that a
weighted oriented graph $D$ is Cohen-Macaulay if $R/I(D)$ is a Cohen-Macaulay
ring.

######  Definition 2.1.

For any homogeneous ideal $I$ in $R$, define the Castelnuovo-Mumford
regularity (or merely regularity) as

$\displaystyle\operatorname{reg}(I)$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{j-i\mid\beta_{i,j}(I)\neq 0\\}$
$\displaystyle=$ $\displaystyle\operatorname{max}\\{j+i\mid
H_{\mathfrak{m}}^{i}(I)_{j}\neq 0\\},$

where $\beta_{i,j}(I)$ is the $(i,j)^{th}$ graded Betti number of $I$ and
$H_{\mathfrak{m}}^{i}(I)_{j}$ denotes the $j^{th}$ graded component of the
$i^{th}$ local cohomology module.

######  Definition 2.2.

Suppose $I$ is a monomial ideal such that $I=J+K$, where
$\mathcal{G}(I)=\mathcal{G}(J)\cup\mathcal{G}(K)$. Then $I=J+K$ is Betti
splitting if

$\displaystyle\beta_{i,j}(I)=\beta_{i,j}(J)+\beta_{i,j}(J)+\beta_{i-1,j}(J\cap
K)\mbox{ for all }i,j\geq 0,$

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

######  Definition 2.3.

Let $m=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$ be a monomial in
$R=k[x_{1},\ldots,x_{n}]$. Then the polarization of $m$ is defined to be the
squarefree monomial

$\mathcal{P}(m)=x_{11}x_{12}\ldots x_{1a_{1}}x_{21}\ldots x_{2a_{2}}\ldots
x_{na_{n}}$

in the polynomial ring $k[x_{ij}:1\leq j\leq a_{i},1\leq i\leq n].$ If
$I\subset R$ is a monomial ideal with
$\mathcal{G}(I)=\\{m_{1},\ldots,m_{u}\\}$ and
$m_{i}=\prod_{j=1}^{n}x_{j}^{a_{ij}}$ where each $a_{ij}\geq 0$ for
$i=1,\ldots,m.$ Then polarization of $I$, denoted by $I^{\mathcal{P}}$, is
defined as:

$I^{\mathcal{P}}=(\mathcal{P}(m_{1}),\ldots,\mathcal{P}(m_{u})),$

which is a squarefree monomial ideal in the polynomial ring
$R^{\mathcal{P}}=k[x_{j1},x_{j2},\ldots,x_{ja_{j}}\mid j=1,\ldots,n]$ where
$a_{j}=\operatorname{max}\\{a_{ij}\mid i=1,\ldots,m\\}$ for any $1\leq j\leq
n.$

The following result conveys that polarization preserves some homological
invariants.

######  Lemma 2.4.

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

  1. (1)

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

  2. (2)

$\operatorname{reg}(I)=\operatorname{reg}(I^{\mathcal{P}})$.

######  Lemma 2.5.

[19, Lemma 2.7] Let $S_{1}=k[x_{1},\ldots,x_{m}]$ and
$S_{2}=k[x_{m+1},\ldots,x_{n}]$ be two polynomial rings, $I\subset S_{1}$ and
$J\subset S_{2}$ be two non zero homogeneous ideals. Then,

  1. (1)

$\operatorname{reg}(I+J)=\operatorname{reg}(I)+\operatorname{reg}(J)-1,$

  2. (2)

$\operatorname{reg}(IJ)=\operatorname{reg}(I)+\operatorname{reg}(J).$

######  Lemma 2.6.

[9, Corollary 12] Let $H$ be a simple hypergraph and $H^{\prime}$ its induced
subhypergraph. Then,

$\operatorname{reg}(H^{\prime})\leq\operatorname{reg}(H).$

The following result follows from Lemma 2.6 and 2.4.

######  Lemma 2.7.

Let $D^{\prime}$ be an induced weighted oriented subgraph of $D$. Then,

$\operatorname{reg}(I(D^{\prime})^{k})\leq\operatorname{reg}(I(D)^{k}).$

Recall the following lemma, which we frequently use in many proofs.

######  Lemma 2.8.

[11, Lemma 1.2] [Regularity lemma] Let $0\rightarrow A\rightarrow B\rightarrow
C\rightarrow 0$ be a short exact sequence finitely generated graded
$R$-modules. Then

  1. (1)

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

  2. (2)

$\operatorname{reg}(A)\leq\operatorname{max}\\{\operatorname{reg}(B),\operatorname{reg}(C)+1\\}$.

  3. (3)

$\operatorname{reg}(C)\leq\operatorname{max}\\{\operatorname{reg}(A)-1,\operatorname{reg}(B)\\}$.

  4. (4)

If $\operatorname{reg}(A)>\operatorname{reg}(C)+1$, then
$\operatorname{reg}(B)=\operatorname{reg}(A)$.

  5. (5)

If $\operatorname{reg}(C)\geq\operatorname{reg}(A)$, then
$\operatorname{reg}(B)=\operatorname{reg}(C)$.

  6. (6)

If $\operatorname{reg}(A)>\operatorname{reg}(B)$, then
$\operatorname{reg}(C)=\operatorname{reg}(A)-1$.

  7. (7)

If $\operatorname{reg}(B)>\operatorname{reg}(A)$, then
$\operatorname{reg}(C)=\operatorname{reg}(B)$.

The following theorem is a combinatorial characterization of Cohen-Macaulay
forests.

######  Theorem 2.9.

[7, Theorem 5] Let $D$ be a weighted oriented forest without isolated
vertices, and let $G$ be its underlying forest. Then the following conditions
are equivalent:

  1. (1)

$D$ is Cohen-Macaulay.

  2. (2)

$I(D)$ is unmixed; that is, all its associated primes have the same height.

  3. (3)

$G$ has a perfect matching
$\\{x_{1},y_{1}\\},\\{x_{2},y_{2}\\},\ldots,\\{x_{r},y_{r}\\}$ so that
$d_{G}(y_{i})=1$ for $i=1,2,\ldots,r$ and $w(x_{i})=1$ if $(x_{i},y_{i})\in
E(D)$.

##  3\. Castelnuovo Mumford Regularity of Cohen-Macaulay weighted oriented
forests

In this section we prove a formula for $\operatorname{reg}(I(D))$. To show
this, we prove some technical lemmas.

######  Lemma 3.1.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any edge $(x_{t},y_{t})$ and $z\in V(D)$ such that
$N_{D}(x_{t})=\\{y_{t},z\\}$ and for all $k\geq 1$,

  1. (1)

$\displaystyle I(D\setminus y_{t})^{k}\bigcap
x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}=zx_{t}y_{t}^{w(y_{t})}I(D\setminus
y_{t})^{k-1}+x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}$,

  2. (2)

$\displaystyle zx_{t}y_{t}^{w(y_{t})}I(D\setminus y_{t})^{k-1}\bigcap
x_{t}y_{t}^{w(y_{t})}I(D\setminus
N_{D}[x_{t}])^{k}=zx_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}$.

###### Proof.

(1) Note that $x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}$ and
$x_{t}y_{t}^{w(y_{t})}zI(D\setminus y_{t})^{k-1}$ are contained in
$I(D\setminus y_{t})^{k}\bigcap x_{t}y_{t}^{w}(y_{t})I(D)^{k-1},$ because
$x_{t}z\in\mathcal{G}(I(D\setminus y_{t})).$ This implies that

$zx_{t}y_{t}^{w(y_{t})}I(D\setminus
y_{t})^{k-1}+x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}\subseteq
I(D\setminus y_{t})^{k}\bigcap x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}.$

Now we prove the reverse inclusion. Let $f\in I(D\setminus y_{t})^{k}\bigcap
x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}$. Then $f=x_{t}y_{t}^{w(y_{t})}f_{1},$ where
$f_{1}\in R$ and $m\mid f,$ for some $m\in\mathcal{G}(I(D\setminus
y_{t})^{k})$. Now We will see the proof in three further cases.
Case 1: Suppose $x_{t}\mid m$. Then $z\mid m$ because
$zx_{t}\in\mathcal{G}(I(D\setminus y_{t})).$ In fact, we have $zx_{t}\mid m$.
Then $m=zx_{t}m_{1}$, for some $m_{1}\in\mathcal{G}(I(D\setminus
y_{t})^{k-1})$. Since $y_{t}^{w(y_{t})}\mid f$, $m\mid f$ and
$y_{t}^{w(y_{t})}\nmid m$, we get that $m\mid x_{t}f_{1}$. This implies that
$zx_{t}m_{1}\mid x_{t}f_{1}$. This gives that $m_{1}\mid f_{1}$. Thus
$f_{1}\in I(D\setminus y_{t})^{k-1}$. This implies that $f\in
zx_{t}y_{t}^{w(y_{t})}I(D\setminus y_{t})^{k-1}.$
Case 2: Suppose $x_{t}\nmid m$ and $z\mid m.$ Then we have
$m\in\mathcal{G}(I(D\setminus\\{x_{t},y_{t}\\})^{k})$. Also, $zu^{w(u)}\mid m$
because $zu^{w(u)}\in\mathcal{G}(I(D\setminus\\{x_{t},y_{t}\\})),$ where $u\in
N_{D}(z)\setminus x_{t}$ . Then $m=zu^{w(u)}m_{2},$ for some
$m_{2}\in\mathcal{G}(I(D\setminus\\{x_{t},y_{t}\\})^{k-1}).$ On the other
hand, we have $f=x_{t}y_{t}^{w(y_{t})}f_{1},$ where $f_{1}\in R$. Thus we have
$m\mid f_{1}$. Therefore,

$f=x_{t}y_{t}^{w(y_{t})}f_{1}=x_{t}y_{t}^{w(y_{t})}zu^{w(u)}m_{2}g_{2},$

for some $g_{2}\in R$. Thus we have

$f\in
zx_{t}y_{t}^{w(y_{t})}u^{w(u)}I(D\setminus\\{x_{t},y_{t}\\})^{k-1}\subseteq
zx_{t}y_{t}^{w(y_{t})}I(D\setminus\\{x_{t},y_{t}\\})^{k-1}\subseteq
zx_{t}y_{t}^{w(y_{t})}I(D\setminus y_{t})^{k-1}.$

Case 3: Suppose $x_{t}\nmid m$ and $z\nmid m.$ Then
$m\in\mathcal{G}(I(D\setminus N_{D}[x_{t}])^{k}).$ Since
$f=x_{t}y_{t}^{w(y_{t})}f_{1},$ we get that $m\mid f_{1}$ because $m\mid f$
and $m\in\mathcal{G}(I(D\setminus N_{D}[x_{t}])^{k}).$ Therefore,

$f=x_{t}y_{t}^{w(y_{t})}f_{1}=x_{t}y_{t}^{w(y_{t})}mg_{3},$

for some $g_{3}\in R.$ This implies that $f\in
x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}.$

(2) Note that $y_{t}^{w(y_{t})}x_{t}zI(D\setminus N_{D}[x_{t}])^{k}\subseteq
y_{t}^{w(y_{t})}x_{t}zI(D\setminus y_{t})^{k-1}\bigcap
x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}.$ Now, we will prove the
other inclusion. Let $f\in y_{t}^{w(y_{t})}x_{t}zI(D\setminus
y_{t})^{k-1}\bigcap x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}.$ Then
$f=y_{t}^{w(y_{t})}x_{t}zf_{1}$, for some $f_{1}\in R$ and
$x_{t}y_{t}^{w(y_{t})}m_{1}\mid f$, for some $m_{1}\in\mathcal{G}(I(D\setminus
N_{D}[x_{t}])^{k})$. This implies that $m_{1}\mid f_{1}.$ This gives that
$f=y_{t}^{w(y_{t})}x_{t}zm_{1}g,$ for some $g\in R.$ Thus $f\in
zx_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}$, as required. ∎

######  Lemma 3.2.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any edge $(x_{t},y_{t})$ and $z\in V(D)$ such that
$N_{D}(x_{t})=\\{y_{t},z\\}$, and for all $k\geq 1$,

  1. (1)

$(I(D\setminus y_{t})^{k}:zx_{t})=I(D\setminus y_{t})^{k-1},$

  2. (2)

$(I(D\setminus y_{t})^{k},x_{t})=(I(D\setminus\\{x_{t},y_{t}\\})^{k},x_{t}),$

  3. (3)

$((I(D\setminus y_{t})^{k}:x_{t}),z)=((I(D\setminus
N_{D}[x_{t}])^{k}:x_{t}),z)=(I(D\setminus N_{D}[x_{t}])^{k},z).$

###### Proof.

Note that $x_{t}$ is a leaf in $D\setminus y_{t}$ and $N_{D\setminus
y_{t}}(x_{t})=\\{z\\}$ as $N_{D}(x_{t})=\\{y_{t},z\\}$. Also, one can write
$N_{D\setminus y_{t}}^{-}(x_{t})=\\{z\\}$ and $w(x_{t})=1,w(z)=1$. Thus the
proof follows from [19, Proposition 3.2]. ∎

######  Proposition 3.3.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then

$\operatorname{reg}(I(D\setminus
y_{t})^{k}\leq\operatorname{max}\\{\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+2,\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+1,\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})\\}.$

###### Proof.

Let us take two exact sequences

(1) $0\rightarrow(I(D\setminus
y_{t})^{k}:x_{t})(-1)\xrightarrow{.x_{t}}I(D\setminus
y_{t})^{k}\rightarrow(I(D\setminus y_{t})^{k},x_{t})\rightarrow 0$ (2)
$0\rightarrow(I(D\setminus y_{t})^{k}:zx_{t})(-1)\xrightarrow{.z}(I(D\setminus
y_{t})^{k}:x_{t})\rightarrow((I(D\setminus y_{t})^{k}:x_{t}),z)\rightarrow 0$

The Lemma 3.2(1), 3.2(2) and 3.2(3) result in

  1. (1)

$\operatorname{reg}((I(D\setminus
y_{t})^{k}:zx_{t}))=\operatorname{reg}(I(D\setminus y_{t})^{k-1}),$

  2. (2)

$\operatorname{reg}(I(D\setminus
y_{t})^{k},x_{t})=\operatorname{reg}((I(D\setminus\\{x_{t},y_{t}\\})^{k},z)),$

  3. (3)

$\operatorname{reg}(((I(D\setminus
y_{t})^{k}:x_{t}),z)=\operatorname{reg}((I(D\setminus N_{D}[x_{t}])^{k},z)),$

respectively. Note that we have
$\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k},x_{t})=\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})$
and $\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k},z)=\operatorname{reg}(I(D\setminus N_{D}[x_{t}])^{k})$ using
Lemma 2.5(1). Now, we apply regularity Lemma 2.8(1) on exact sequence (2) to
get

$\displaystyle\operatorname{reg}(I(D\setminus y_{t})^{k}:x_{t})$
$\displaystyle\leq\operatorname{max}\\{\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+1,\operatorname{reg}(I(D\setminus N_{D}[x_{t}])^{k},z)\\}$
$\displaystyle=$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+1,\operatorname{reg}(I(D\setminus N_{D}[x_{t}])^{k})\\}.$

Also, we apply regularity lemma 2.8(1) on exact sequence (1) to get

$\displaystyle\operatorname{reg}(I(D\setminus y_{t})^{k}\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}((I(D\setminus
y_{t})^{k}):x_{t})+1,\operatorname{reg}((I(D\setminus\\{x_{t},y_{t}\\})^{k},x_{t}))\\}$
$\displaystyle=$
$\displaystyle\operatorname{max}\\{\operatorname{reg}((I(D\setminus
y_{t})^{k}):x_{t})+1,\operatorname{reg}((I(D\setminus\\{x_{t},y_{t}\\})^{k})\\}.$

Thus

$\operatorname{reg}(I(D\setminus
y_{t})^{k}\leq\operatorname{max}\\{\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+2,\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+1,\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})\\}.$

This proves the proposition. ∎

Notation : We fix the following notation for any weighted oriented graph $D$:

$\displaystyle\Theta(k,D)=$
$\displaystyle\operatorname{max}\Bigg{\\{}(\operatorname{max}\\{w(y_{i_{j}})\\}+1)(k-1)+\sum_{j=1}^{s}w(y_{i_{j}})+1:\mbox{
none of the edges }\\{x_{i_{j}},y_{i_{j}}\\}$ $\displaystyle\mbox{ are
adjacent },y_{i_{j}}\in V(D)\Bigg{\\}},\mbox{ for all }k\geq 1,\mbox{ and }$
$\displaystyle\Theta(0,D)$ $\displaystyle=0.$

######  Lemma 3.4.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any edge $(x_{t},y_{t})$ and $z\in V(D)$ such that
$N_{D}(x_{t})=\\{y_{t},z\\}$, we have

$\displaystyle\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D).$

###### Proof.

Suppose $\Theta(k-1,D)=(k-2)(w(y_{j_{i}})+1)+\sum_{n=1}^{n=s}w(y_{j_{n}})+1$
for some $y_{i}^{\prime}s$ and
$w(y_{j_{i}})=\operatorname{max}\\{y_{j_{1}},\cdots,y_{j_{s}}\\}$. Then we
have

$\displaystyle\Theta(k-1,D)+w(y_{t})+1=$
$\displaystyle(k-1)(w(y_{j_{i}}))+1)+\sum_{n=1}^{s}w(y_{j_{n}})+1-(w(y_{j_{i}})+1)+(w(y_{t})+1)$
$\displaystyle=$
$\displaystyle(k-1)(w(y_{j_{i}}))+1)+\sum_{n=1}^{s}w(y_{j_{n}})+1+w(y_{t})-w(y_{j_{i}})$
$\displaystyle\leq$ $\displaystyle\Theta(k,D)+w(y_{t})-w(y_{j_{i}}).$

Suppose $y_{t}\in\\{y_{j_{1}},\ldots,y_{j_{s}}\\}$, then we have
$\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D).$
Now suppose $y_{t}\not\in\\{y_{j_{1}},\ldots,y_{j_{s}}\\}$. Then
$y_{j}\in\\{y_{j_{1}},\ldots,y_{j_{s}}\\}$ and

$\sum_{n=1,i_{n}\neq
j}^{s}w(y_{j_{n}})+1+w(y_{j})\leq\Theta(1,D\setminus\\{N_{D}[x_{t}]\cup
N_{D}[z]\\})+w(y_{j})\leq\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{j})$

for $(z,y_{j})\in E(D)$ as $N_{D}(x_{t})=\\{z,y_{t}\\}$. If possible suppose
$w(y_{t})>w(y_{j_{i}})$, then we have

$\displaystyle\Theta(k-1,D)=$
$\displaystyle(k-2)(w(y_{j_{i}})+1)+\sum_{n=1,i_{n}\neq
j}^{s}w(y_{j_{n}})+1+w(y_{j})$ $\displaystyle\leq$
$\displaystyle(k-2)(w(y_{j_{i}})+1)+\Theta(1,D\setminus\\{N_{D}[x_{t}]\cup
N_{D}[z]\\})+w(y_{j})$ $\displaystyle<$
$\displaystyle(k-2)(w(y_{t})+1)+\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t})\mbox{ (since }w(y_{t})>w(y_{j_{i}})\geq w(y_{j}))$
$\displaystyle\leq$ $\displaystyle\Theta(k-1,D).$

where last inequality holds because set of all non-adjacent pairs
$\\{x_{i_{j}},y_{i_{j}}\\}$ in $D\setminus N_{D}[x_{t}]$ are also non-adjacent
to $\\{x_{t},y_{t}\\}$ in $D$. Therefore we get contradiction that
$y_{t}\not\in\\{y_{j_{1}},\ldots,y_{j_{s}}\\}$. Hence $w(y_{t})\leq
w(y_{j_{i}}).$ This implies that

$\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D).$

∎

######  Lemma 3.5.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any edge $(x_{t},y_{t})$ and $z\in V(D)$ such that
$N_{D}(x_{t})=\\{y_{t},z\\}$, we have

$\displaystyle\Theta(k,D)=\operatorname{max}\\{\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k-1,D)+w(y_{t})+1\\}.$

###### Proof.

Note that $\Theta(k,D\setminus\\{x_{t},y_{t}\\})\leq\Theta(k,D)$. Since set of
all non-adjacent pairs $\\{x_{i_{j}},y_{i_{j}}\\}$ in $D\setminus
N_{D}[x_{t}]$ are also non-adjacent to $\\{x_{t},y_{t}\\}$ in $D$, then

$\Theta(k,D\setminus N_{D}[x_{t}])+w(y_{t})\leq\Theta(k,D).$

Also, from Lemma 3.4, we have

$\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D).$

This implies that

$\displaystyle\operatorname{max}\\{\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k-1,D)+w(y_{t})+1\\}\leq\Theta(k,D).$

Now we show the other inequality. Suppose
$\Theta(k,D)=(w(y_{l}))+1)(k-1)+\sum_{j=1}^{s}w(y_{i_{j}})+1$, where
$w(y_{l})=\operatorname{max}\\{w(y_{i_{1}}),\ldots,w(y_{i_{s}})\\}$ for some
$l$. If none of the $y_{i_{j}}$ is equal to $y_{t}$, then

$(k-1)(w(y_{l}))+1)+\sum_{j=1}^{s}w(y_{i_{j}})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\}).$

If $y_{i_{j_{0}}}=y_{t}$ for some $i_{j_{0}}$ and $y_{l}\neq y_{t}$, then from
the non-adjacency of edges $\\{x_{i_{j}},y_{i_{j}}\\}$ we get

$\displaystyle(k-1)(w(y_{l}))+1)+\sum_{j=1}^{s}w(y_{i_{j}})+1$
$\displaystyle=(k-1)(w(y_{l})+1)+\sum_{j=1,j\neq
t}^{s}w(y_{i_{j}})+1+w(y_{t})$ $\displaystyle\leq\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}).$

Now assume $y_{i_{j_{0}}}=y_{l}=y_{t}$. Then

$\displaystyle(w(y_{l})+1)(k-1)+\sum_{j=1}^{s}w(y_{i_{j}})+1$
$\displaystyle=(k-1)(w(y_{t})+1)+\sum_{j=1,j\neq t}^{s}w(y_{i_{j}})+1$
$\displaystyle=(k-2)(w(y_{t})+1)+\sum_{j=1,j\neq
t}^{s}w(y_{i_{j}})+1+w(y_{t})+1$ $\displaystyle\leq\Theta(k-1,D)+w(y_{t})+1.$

Thus we get the inequality

$\displaystyle\Theta(k,D)\leq\operatorname{max}\\{\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k-1,D)+w(y_{t})+1\\}.$

This proves the lemma. ∎

Now we prove the main result of this section.

######  Theorem 3.6.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then

$\displaystyle\operatorname{reg}(I(D))=\Theta(1,D)=$
$\displaystyle\operatorname{max}\Bigg{\\{}\sum_{j=1}^{s}w(y_{i_{j}})+1~{}:\mbox{
none of the edges }\\{x_{i_{j}},y_{i_{j}}\\}$ $\displaystyle\mbox{ are
adjacent },y_{i_{j}}\in V(D)\Bigg{\\}}.$

###### Proof.

Let $D$ be a weighted oriented unmixed forest with the perfect matching
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$. Note that $r=\frac{\mid
V(D)\mid}{2}$. We prove the theorem by induction on $r$. Let $r=2$. We have
that $I(D)=(x_{1}x_{2},x_{1}y_{1}^{w(y_{1})},x_{2}y_{2}^{w(y_{2})})$. Write
$I(D)=J+K$, where $J=(x_{1}x_{2},x_{1}y_{1}^{w(y_{1})})$,
$K=(x_{2}y_{2}^{w(y_{2})})$. Note that $J\cap K=(x_{1}x_{2}y_{2}^{w(y_{2})})$.
One can easily see that
$\operatorname{reg}(J)=w(y_{1})+1,\operatorname{reg}(K)=w(y_{2})+1$ and
$\operatorname{reg}(J\cap K)=w(y_{2})+2$. Now applying the regularity lemma to
the short exact sequence,

(3) $0\rightarrow J\cap K\rightarrow J\oplus K\rightarrow J+K\rightarrow 0,$

we get that

$\displaystyle\operatorname{reg}(I(D))$ $\displaystyle=$
$\displaystyle\operatorname{reg}(J+K)$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(J\cap
K)-1,\operatorname{reg}(J\oplus K)\\}$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{w(y_{1})+1,w(y_{2})+1\\}.$

The other inequality is true because $I(D)$ has minimal generators of degrees
$w(y_{1})+1$ and $w(y_{2})+1$. Thus
$\operatorname{reg}(I(D))=\operatorname{max}\\{w(y_{1})+1,w(y_{2})+1\\}$.
Therefore the theorem is true for $r=2$. Assume $r\geq 3$. Choose an edge
$(x_{t},y_{t})$ such that $N_{D}(x_{t})=\\{z,y_{t}\\}$. Such a choice of an
edge always exists in $D$. Write $I(D)=J+K$, where $J=I(D\setminus y_{t})$ and
$K=(x_{t}y_{t}^{w(y_{t})})$. Note that $\operatorname{reg}(K)=w(y_{t})+1$. By
induction hypothesis apply on $D\setminus N_{D}[x_{t}]$ and
$D\setminus\\{x_{t},y_{t}\\}$ we get that

$\displaystyle\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\}))=\Theta(1,D\setminus\\{x_{t},y_{t}\\})\text{
and }\operatorname{reg}(I(D\setminus N_{D}[x_{t}]))=\Theta(1,D\setminus
N_{D}[x_{t}]).$

From Lemma 3.1(1) we have $J\cap K=x_{t}y_{t}^{w(y_{t})}[I(D\setminus
N_{D}[x_{t}])+(z)].$ This implies that

$\displaystyle\operatorname{reg}(J\cap K)=$
$\displaystyle\operatorname{reg}((z)+I(D\setminus N_{D}[x_{t}]))+w(y_{t})+1$
$\displaystyle=$ $\displaystyle\operatorname{reg}(I(D\setminus
N_{D}[x_{t}]))+1+w(y_{t})+1-1\mbox{ ( by Lemma \ref{lem5} (1) })$
$\displaystyle=$ $\displaystyle\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t})+1.$

Now from Proposition 3.3 with $k=1$ and $\operatorname{reg}(I(D\setminus
N_{D}[x_{t}]))\geq 2$ we get that
$\operatorname{reg}(I(D\setminus
y_{t})\leq\operatorname{max}\\{\operatorname{reg}(I(D\setminus
N_{D}[x_{t}]))+1,\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\}))\\}.$ Thus
we have

(4) $\displaystyle\operatorname{reg}(J)=\operatorname{reg}(I(D\setminus
y_{t})\leq$ $\displaystyle\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+1,\Theta(1,D\setminus\\{x_{t},y_{t}\\})\\}.$

First, we will show that $\operatorname{reg}(I(D))\leq\Theta(1,D)$. Consider
the exact sequence (3) and apply Lemma 2.8(3) we get that

$\displaystyle\operatorname{reg}(I(D))=$
$\displaystyle\operatorname{reg}(J+K)$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(J\cap
K)-1,\operatorname{reg}(J\oplus K)\\}$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t}),\operatorname{reg}(J),w(y_{t})+1\\}$
$\displaystyle\leq$ $\displaystyle\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(1,D\setminus
N_{D}[x_{t}])+1,\Theta(1,D\setminus\\{x_{t},y_{t}\\})\\}$ $\displaystyle\hskip
256.0748pt(\mbox{ from equation \eqref{eq6} })$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(1,D\setminus\\{x_{t},y_{t}\\})\\}$
$\displaystyle=$ $\displaystyle\Theta(1,D)\mbox{ ( from Lemma \ref{lm3.6} }).$

This implies that $\operatorname{reg}(I(D))\leq\Theta(1,D).$ Now, we will show
that $\Theta(1,D)\leq\operatorname{reg}(I(D))$ in two cases.
Case $1$: Suppose $\operatorname{reg}(J\oplus K)\geq\operatorname{reg}(J\cap
K)$. Then

$\displaystyle\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t})=$
$\displaystyle\operatorname{reg}(I(D\setminus N_{D}[x_{t}]))+w(y_{t})$
$\displaystyle=$ $\displaystyle\operatorname{reg}(J\cap K)$
$\displaystyle\leq$ $\displaystyle\operatorname{reg}(J\oplus K)$
$\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(J),\operatorname{reg}(K)\\}$
$\displaystyle\leq$ $\displaystyle\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+1,\Theta(1,D\setminus\\{x_{t},y_{t}\\}),2\\}$
$\displaystyle\leq$
$\displaystyle\Theta(1,D\setminus\\{x_{t},y_{t}\\})\leq\operatorname{reg}(I(D)),$

where the last inequality holds by the virtue of
$\Theta(1,D\setminus\\{x_{t},y_{t}\\})=\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})$
and Lemma 2.7. Thus from Lemma 3.5, we have

$\Theta(1,D)=\operatorname{max}\\{\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(1,D\setminus\\{x_{t},y_{t}\\})\\}\leq\operatorname{reg}(I(D)).$

This implies that $\Theta(1,D)\leq\operatorname{reg}(I(D))$.
Case $2$: Suppose $\operatorname{reg}(J\oplus K)<\operatorname{reg}(J\cap K)$.
Then by applying Lemma 2.8(6) on exact sequence we get that

$\operatorname{reg}(I(D))=\operatorname{reg}(J+K)=\operatorname{reg}(J\cap
K)-1=\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t}).$

Since $D\setminus\\{x_{t},y_{t}\\}$ is an induced subgraph of a $D$, then from
Lemma 2.7 we have

$\Theta(1,D\setminus\\{x_{t},y_{t}\\})=\operatorname{reg}(1,D\setminus\\{x_{t},y_{t}\\})\leq\operatorname{reg}(I(D)).$

Thus, from Lemma 3.5 we have

$\Theta(1,D)=\operatorname{max}\\{\Theta(1,D\setminus\\{x_{t},y_{t}\\}),\Theta(1,D\setminus
N_{D}[x_{t}]))+w_{t}\\}\leq\operatorname{reg}(I(D).$

This finishes the proof. ∎

######  Example 3.7.

Let $D$ be a Cohen-Macaulay weighted oriented graph with all leaves are sinks
as below.

Figure 1. Cohen-Macaulay weighted oriented forests

The weight of vertices are $w(x_{i})=1$, for $1\leq i\leq 6$, $w(y_{1})=5$,
$w(y_{2})=7$, $w(y_{3})=3$, $w(y_{4})=4$, $w(y_{5})=6$, $w(y_{6})=9$.The edge
ideal of $D$ is

$I(D)=(x_{1}x_{2},x_{2}x_{3},x_{2}x_{4},x_{3}x_{6},x_{2}x_{5},x_{1}y_{1}^{5},x_{2}y_{2}^{7},x_{3}y_{3}^{3},x_{4}y_{4}^{4},x_{5}y_{5}^{6},x_{6}y_{6}^{9}).$

By the Theorem (3.6), we have

$\displaystyle\operatorname{reg}(I(D))$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{w(y_{1})+w(y_{3})+w(y_{4})+w(y_{5})+1,w(y_{1})+w(y_{4})+w(y_{5})+w(y_{6})+1,$
$\displaystyle w(y_{2})+w(y_{6})+1\\}$ $\displaystyle=$ $\displaystyle 25.$

Also one can check that $\operatorname{reg}(I(D))=25$ by using Macaulay$2$,
[8].

##  4\. Castelnuovo-Mumford regularity of power of edge ideal of Cohen-
Macaulay weighted oriented forests

In this section we prove a formula for $\operatorname{reg}(I(D)^{k})$. To
prove that we need some lemmas which are below.

######  Lemma 4.1.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$ such that $y_{1},\cdots,y_{r}$ are sinks.
Suppose $(x_{t},y_{t})\in E(D)$ such that $N_{D}(x_{t})=\\{y_{t},z\\}$
satisfying

$\operatorname{max}\\{\Theta(k-1,D),\Theta(k,D\setminus
N_{D}[x_{t}])\\}\leq\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+1.$

Then

$\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\}).$

###### Proof.

Suppose
$\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})=(k-2)(w(y_{i_{u}})+1)+\sum_{n=1}^{s}w(y_{i_{n}})+1$,
for some $y_{i}^{\prime}s$ and
$w(y_{i_{u}})=\operatorname{max}\\{w(y_{i_{1}}),\ldots,w(y_{i_{s}})\\}$. Then
we have

$\displaystyle\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+1=$
$\displaystyle(k-2)(w(y_{i_{u}})+1)+\sum_{n=1}^{s}w(y_{i_{n}})+1+(w(y_{t})+1)$
$\displaystyle=$
$\displaystyle(k-1)(w(y_{i_{u}})+1)+\sum_{n=1}^{s}w(y_{i_{n}})+1+w(y_{t})-w(y_{i_{u}})$
$\displaystyle\leq$ $\displaystyle\Theta(k,D)+w(y_{t})-w(y_{i_{u}}).$

If possible, suppose $y_{j}\not\in\\{y_{i_{1}},\ldots,y_{i_{s}}\\}$ for
$(z,y_{j})\in E(D)$ as $N_{D}(x_{t})=\\{z,y_{t}\\}$, then

$\displaystyle\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})=(k-2)(w(y_{i_{u}})+1)+\sum_{n=1,i_{n}\neq
j}^{s}w(y_{i_{n}})+1\leq\Theta(k-1,D\setminus N_{D}[x_{t}]).$

Thus from $\Theta(k-1,D\setminus N_{D}[x_{t}])+2\leq\Theta(k,D\setminus
N_{D}[x_{t}])$

$\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+1\leq\Theta(k-1,D\setminus
N_{D}[x_{t}])+1<\Theta(k,D\setminus N_{D}[x_{t}])$

which contradict our assumption $\Theta(k,D\setminus
N_{D}[x_{t}])\leq\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+1.$
Therefore $y_{j}\in\\{y_{i_{1}},\ldots,y_{i_{s}}\\}$. This implies that

$\sum_{n=1,i_{n}\neq
j}^{s}w(y_{i_{n}})+1+w(y_{j})\leq\Theta(1,D\setminus\\{N_{D}[x_{t}]\cup
N_{D}[z]\\})+w(y_{j})\leq\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{j})$

for $(z,y_{j})\in E(D)$ as $N_{D}(x_{t})=\\{z,y_{t}\\}$. If possible suppose
$w(y_{t})>w(y_{i_{u}})$, then we have

$\displaystyle\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+1=$
$\displaystyle\>(k-2)(w(y_{i_{u}})+1)+\sum_{n=1,i_{n}\neq
j}^{s}w(y_{i_{n}})+1+w(y_{j})+1$ $\displaystyle<$
$\displaystyle\>(k-2)(w(y_{t})+1)+\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t})$
$\displaystyle\hskip 147.95424pt{\mbox{ (since }w(y_{t})>w(y_{i_{u}})\geq
w(y_{j}))}$ $\displaystyle\leq$ $\displaystyle\>\Theta(k-1,D)$

where the last inequality holds because set of all non-adjacent pairs
$\\{x_{i_{j}},y_{i_{j}}\\}$ in $D\setminus N_{D}[x_{t}]$ are also non-adjacent
to $\\{x_{t},y_{t}\\}$ in $D$. Therefore we have
$\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+1<\Theta(k,D)$ which
contradict our assumption that
$\Theta(k,D)\leq\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+1.$ Hence
$w(y_{t})\leq w(y_{i_{u}}).$ This implies that

$\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\}).$

∎

######  Lemma 4.2.

Let $D$ be a weighted oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\cdots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$ such that $y_{1},\ldots,y_{r}$ are sinks.
Suppose $(x_{t},y_{t})\in E(D)$ such that $N_{D}(x_{t})=\\{y_{t},z\\}$
satisfying

(#)
$\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})\\}<\Theta(k-1,D)+w(y_{t})+1$

for some $k\geq 1$. Then

$\Theta(k-1,D)+w(y_{t})+1=(k-1)(w(y_{t})+1)+\Theta(1,D\setminus
N_{D}[x_{t}])+w(y_{t}).$

###### Proof.

Suppose $\Theta(k-1,D)=(k-2)(w(y_{j_{i}})+1)+\sum_{n=1}^{n=s}w(y_{j_{n}})+1$
for some $y_{i}^{\prime}s$ and
$w(y_{j_{i}})=\operatorname{max}\\{y_{1},\ldots,y_{r}\\}$. If possible,
suppose $y_{t}\not\in\\{y_{j_{1}},\ldots,y_{j_{s}}\\}$. Then
$\Theta(k-1,D)=\Theta(k-1,D\setminus\\{x_{t},y_{t}\\}).$ Note that from
equation (# ‣ 4.2) we get that $\Theta(k,D\setminus
N_{D}[x_{t}])\leq\Theta(k,D)$. This implies that

$\operatorname{max}\\{\Theta(k-1,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])\\}<\Theta(k,D\setminus\\{x_{t},y_{t}\\})+1.$

Thus from Lemma 4.1, we get that
$\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\})$ which
contradicts equation (# ‣ 4.2). Now, if possible, suppose $y_{t}\neq
y_{j_{i}}$. Then $w(y_{t})\leq w(y_{j_{i}})$. Consider,

$\displaystyle\Theta(k-1,D)+w(y_{t})+1=$
$\displaystyle\>(k-2)(w(y_{j_{i}})+1)+\sum_{n=1}^{n=s}w(y_{j_{n}})+1+(w(y_{t})+1)$
$\displaystyle\leq$
$\displaystyle\>(k-2)(w(y_{j_{i}})+1)+\sum_{n=1}^{n=s}w(y_{j_{n}})+1+(w(y_{j_{i}})+1)$
$\displaystyle=$ $\displaystyle\>(k-1)(w(y_{j_{i}})+1)+\sum_{n=1,j_{n}\neq
t}^{n=s}w(y_{j_{n}})+1+w(y_{t})$ $\displaystyle\leq$
$\displaystyle\>\Theta(k,D\setminus N_{D}[x_{t}])+w(y_{t})$

which contradict equation (# ‣ 4.2). Therefore $y_{t}=y_{j_{i}}.$ Thus

$\displaystyle\Theta(k-1,D)+w(y_{t})+1=$
$\displaystyle(k-2)(w(y_{t})+1)+\sum_{n=1}^{n=s}w(y_{j_{n}})+1+(w(y_{t})+1)$
$\displaystyle=$ $\displaystyle(k-1)(w(y_{t})+1)+\sum_{n=1,j_{n}\neq
t}^{n=s}w(y_{j_{n}})+1+w(y_{t})$ $\displaystyle=$
$\displaystyle(k-1)(w(y_{t})+1)+\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t}).$

This finishes the proof. ∎

We prove the main result of this section.

######  Theorem 4.3.

Let $D$ be an oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any $k\geq 1$,

$\displaystyle\operatorname{reg}(I(D)^{k})=\Theta(k,D)=$
$\displaystyle\operatorname{max}\Bigg{\\{}(k-1)(\operatorname{max}\\{w(y_{i_{j}})\\}+1)+\sum_{j=1}^{s}w(y_{i_{j}})+1:\mbox{
none of the}$ $\displaystyle\mbox{ edges }\\{x_{i_{j}},y_{i_{j}}\\}\mbox{ are
adjacent },y_{i_{j}}\in V(D)\Bigg{\\}},\mbox{ for all }k\geq 1.$

###### Proof.

Let $D$ be a weighted oriented unmixed forest with the perfect matching
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$. We prove the theorem by
induction on $k$. If $k=1$, then the theorem is true by Theorem (3.6). Assume
$k\geq 2$. Choose an edge $(x_{t},y_{t})$ such that
$N_{D}(x_{t})=\\{z,y_{t}\\}$. Such a choice of an edge always exists in $D$.
Write

$\displaystyle I(D)^{k}$ $\displaystyle=$ $\displaystyle(I(D\setminus
y_{t})+(x_{t}y_{t}^{w(y_{t})}))^{k}$ $\displaystyle=$
$\displaystyle\sum_{i+j=k,i,j\geq 0,}I(D\setminus
y_{t})^{i}(x_{t}y_{t}^{w(y_{t})})^{j},$ $\displaystyle=$ $\displaystyle
I(D\setminus y_{t})^{k}+x_{t}y_{t}^{w(y_{t})}\sum_{i+j=k-1,i,j\geq
0}I(D\setminus y_{t})^{i}(x_{t}y_{t}^{w(y_{t})})^{j},$ $\displaystyle=$
$\displaystyle I(D\setminus y_{t})^{k}+x_{t}y_{t}^{w(y_{t})}I(D)^{k-1},$
$\displaystyle=$ $\displaystyle J+L,$

where $J=I(D\setminus y_{t})^{k}$ and $L=x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}.$ To
show the $\Theta(k,D)=\operatorname{reg}(I(D)^{k})$, we prove by induction on
$\frac{\mid V(D)\mid}{2}$. Suppose $\frac{\mid V(D)\mid}{2}=1$. Then $D$ has
only one edge $\\{x_{t},y_{t}\\}.$ Then
$\operatorname{reg}(I(D)^{k})=k(w(y_{t})+1)$ for all $k\geq 1$. Assume
$\frac{\mid V(D)\mid}{2}\geq 2$ and the induction hypothesis. By induction
hypothesis we have

$\displaystyle\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})=\Theta(k,D\setminus\\{x_{t},y_{t}\\})\text{
and }\operatorname{reg}(I(D\setminus N_{D}[x_{t}])^{k})=\Theta(k,D\setminus
N_{D}[x_{t}]).$

By Proposition 3.3 we have

$\displaystyle\operatorname{reg}(I(D\setminus y_{t})^{k}\leq$
$\displaystyle\>\operatorname{max}\\{\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+2,\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+1,\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})\\}$
$\displaystyle\leq$
$\displaystyle\>\operatorname{max}\\{\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k}),\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k-1}+2,\ldots,\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})$
$\displaystyle\>+2(k-1),\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+1,\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k-1}+1+2,\ldots,$
$\displaystyle\>\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])+1+2(k-1)\\}\mbox{ (using Proposition \ref{prop3} recursively })$
$\displaystyle=$
$\displaystyle\>\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+2,\ldots,\Theta(1,D\setminus\\{x_{t},y_{t}\\})$
$\displaystyle\>+2(k-1),\Theta(k,D\setminus
N_{D}[x_{t}])+1,\Theta(k-1,D\setminus N_{D}[x_{t}])+1+2,\ldots,$
$\displaystyle\>\Theta(1,D\setminus N_{D}[x_{t}])+1+2(k-1)\\}$

Thus
$\Theta(k,D\setminus\\{x_{t},y_{t}\\})-\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})\geq
2$, $\Theta(k,D\setminus N_{D}[x_{t}])-\Theta(k-1,D\setminus N_{D}[x_{t}])\geq
2$ give that

(5) $\displaystyle\operatorname{reg}(J)=\operatorname{reg}(I(D\setminus
y_{t})^{k}\leq$ $\displaystyle\operatorname{max}\\{\Theta(k,D\setminus
N_{D}[x_{t}])+1,\Theta(k,D\setminus\\{x_{t},y_{t}\\})\\}.$

Also we have
$\operatorname{reg}(L)=\operatorname{reg}(x_{t}y_{t}^{w(y_{t})}I(D)^{k-1})=\operatorname{reg}(I(D)^{k-1})+w(y_{t})+1$.
Then by induction

(6) $\displaystyle\operatorname{reg}(L)=\Theta(k-1,D)+w(y_{t})+1.$

Recall that $I(D)^{k}=J+L$, where $J=I(D\setminus y_{t})^{k}$ and
$L=x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}.$ Then we have

$\displaystyle J\cap L$ $\displaystyle=$ $\displaystyle I(D\setminus
y_{t})^{k}\cap x_{t}y_{t}^{w(y_{t})}I(D)^{k-1}$ $\displaystyle=$
$\displaystyle zx_{t}y_{t}^{w(y_{t})}I(D\setminus
y_{t})^{k-1}+x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}\mbox{ ( By
Lemma \ref{lem2}(1) )}$ $\displaystyle=$ $\displaystyle M+N,$

where $M=zx_{t}y_{t}^{w(y_{t})}I(D\setminus y)^{k-1}$ and
$N=x_{t}y_{t}^{w(y_{t})}I(D\setminus N_{D}[x_{t}])^{k}.$ Note that

$\operatorname{reg}(N)=\operatorname{reg}(x_{t}y_{t}^{w(y_{t})}I(D\setminus
N_{D}[x_{t}])^{k})=\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+w(y_{t})+1$

Thus by the induction hypothesis, we have

(7) $\displaystyle\operatorname{reg}(N)=\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})+1.$

Now,

$\displaystyle\operatorname{reg}(M)=$
$\displaystyle\operatorname{reg}(zx_{t}y_{t}^{w(y_{t})}I(D\setminus
y_{t})^{k-1})$ $\displaystyle=$ $\displaystyle\operatorname{reg}(I(D\setminus
y_{t})^{k-1})+w(y_{t})+2$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\Theta(k-1,D\setminus\\{x_{t},y_{t}\\}),\Theta(k-1,D\setminus
N_{D}[x_{t}])+1\\}+w(y_{t})+2$ $\displaystyle\hskip 284.52756pt\mbox{ ( from
equation \eqref{eq4.1} )}.$

Thus $\Theta(k,D\setminus N_{D}[x_{t}])-\Theta(k-1,D\setminus
N_{D}[x_{t}])\geq 2$ and equation(7) give

(8)
$\displaystyle\operatorname{reg}(M)\leq\operatorname{max}\\{\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})+w(y_{t})+2,\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})+1\\}.$

Since $\Theta(k-1,D\setminus\\{x_{t},y_{t}\\})\leq\Theta(k-1,D)$ and
$\Theta(k-1,D\setminus N_{D}[x_{t}])+1\leq\Theta(k-1,D)$, then equation (8)
implies that

(9) $\displaystyle\operatorname{reg}(M)\leq\Theta(k-1,D)+w(y_{t})+2.$

The Lemma 3.1 gives $M\cap N=zx_{t}y_{t}^{w(y_{t})}I(D\setminus
N_{D}[x_{t}])^{k}$. Then we have

$\operatorname{reg}(M\cap
N)=\operatorname{reg}(zx_{t}y_{t}^{w(y_{t})}I(D\setminus
N_{D}[x_{t}])^{k})=\operatorname{reg}(I(D\setminus
N_{D}[x_{t}])^{k})+w(y_{t})+2.$ (10) $\displaystyle\operatorname{reg}(M\cap
N)=\Theta(k,D\setminus N_{D}[x_{t}])+w(y_{t})+2.$

Now consider two short exact sequences,

(11) $0\rightarrow M\cap N\rightarrow M\oplus N\rightarrow M+N\rightarrow 0,$
(12) $0\rightarrow J\cap L\rightarrow J\oplus L\rightarrow J+L\rightarrow 0,$

and apply the Lemma 2.8(3), we get that

$\displaystyle\operatorname{reg}(I(D)^{k})=$
$\displaystyle\operatorname{reg}(J+L)$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(J\cap
L)-1,\operatorname{reg}(J\oplus L)\\}$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(M+N)-1,\operatorname{reg}(J),\operatorname{reg}(L)\\}\mbox{
(because }J\cap L=M+N)$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(M\cap
N)-2,\operatorname{reg}(M\oplus
N)-1,\operatorname{reg}(J),\operatorname{reg}(L)\\}$ $\displaystyle=$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(M\cap
N)-2,\operatorname{reg}(M)-1,\operatorname{reg}(N)-1,\operatorname{reg}(J),\operatorname{reg}(L)\\}$

From equation (5),(6), (7), (9) and (10) we get that

$\displaystyle\operatorname{reg}(I(D)^{k})=$
$\displaystyle\operatorname{reg}(J+L)$ $\displaystyle\leq$
$\displaystyle\operatorname{max}\\{\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(k-1,D)+w(y_{t})+1,\Theta(k,D\setminus\\{x_{t},y_{t}\\}\\}$
$\displaystyle=$ $\displaystyle\>\Theta(k,D)\mbox{ ( from Lemma \ref{lm3.6}
}).$

This implies that $\operatorname{reg}(I(D)^{k})\leq\Theta(k,D).$ To show that
$\Theta(k,D)\leq\operatorname{reg}(I(D)^{k}),$ it suffices to show that

  1. (i)

$\Theta(k,D\setminus\\{x_{t},y_{t}\\})\leq\operatorname{reg}(I(D)^{k})$,

  2. (ii)

$\Theta(k-1,D)+w(y_{t})+1\leq\operatorname{reg}(I(D)^{k}),$

  3. (iii)

$\Theta(k,D\setminus N_{D}[x_{t}])+w(y_{t})\leq\operatorname{reg}(I(D)^{k}),$

by using Lemma 3.5. Note that (1) is always true using Lemma 2.7 as
$D\setminus\\{x_{t},y_{t}\\}$ is an induced subgraph of $D$. To show (2), we
will take three possible cases.
Case 1: Suppose $\operatorname{reg}(J\oplus L)>\operatorname{reg}(J\cap L).$
Then applying regularity Lemma 2.8(7) on exact sequence (12) we get
$\operatorname{reg}(I(D)^{k})=\operatorname{reg}(J\oplus L).$ Thus from
equation (6) and $\operatorname{reg}(L)\leq\operatorname{reg}(J\oplus L)$ we
get that $\Theta(k-1,D)+w(y_{t})+1\leq\operatorname{reg}(I(D)^{k}).$
Case 2: Suppose $\operatorname{reg}(J\oplus L)<\operatorname{reg}(J\cap L).$
Then applying regularity Lemma 2.8(6) on exact sequence (12) we get that
$\operatorname{reg}(I(D)^{k})=\operatorname{reg}(J\cap L)-1$. Thus from
equation (6) and $\operatorname{reg}(J\oplus L)\leq\operatorname{reg}(J\cap
L)-1$ we get that $\Theta(k-1,D)+w(y_{t})+1\leq\operatorname{reg}(I(D)^{k}).$
Case 3: Suppose $\operatorname{reg}(J\oplus L)=\operatorname{reg}(J\cap L).$
Then applying regularity Lemma 2.8(3) on exact sequence (12) we get that

$\displaystyle\operatorname{reg}(I(D)^{k})\leq$
$\displaystyle\operatorname{max}\\{\operatorname{reg}(J\cap
L)-1,\operatorname{reg}(J\oplus L)\\}=\operatorname{reg}(J\oplus L)$

Thus from equations (5) and (6) we have

(13)
$\displaystyle\operatorname{reg}(I(D)^{k})\leq\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+1,\Theta(k-1,D)+w(y_{t})+1\\}.$

If $\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\}),$ then
we have

$\Theta(k-1,D)+w(y_{t})+1\leq\Theta(k,D\setminus\\{x_{t},y_{t}\\})=\operatorname{reg}(I(D\setminus\\{x_{t},y_{t}\\})^{k})\leq\operatorname{reg}(I(D)^{k})$

because $D\setminus\\{x_{t},y_{t}\\}$ is an induced subgraph of $D$.
Now suppose $\Theta(k,D\setminus\\{x_{t},y_{t}\\})<\Theta(k-1,D)+w(y_{t})+1$.
Then we will take two subcases. Subcase 1 : Suppose $\operatorname{reg}(M\cap
N)\leq\operatorname{reg}(M\oplus N).$ Then applying regularity Lemma 2.8(3) on
exact sequence (11) we get that

$\displaystyle\operatorname{reg}(M+N)$
$\displaystyle\leq\operatorname{max}\\{\operatorname{reg}(M\cap
N)-1,\operatorname{reg}(M\oplus N\\},$
$\displaystyle=\operatorname{reg}(M\oplus N)=\operatorname{reg}(M)\mbox{ (
since }\operatorname{reg}(M\cap N)>\operatorname{reg}(N)).$

Hence,

$\operatorname{reg}(J\oplus L)=\operatorname{reg}(J\cap
L)=\operatorname{reg}(M+N)\leq\operatorname{reg}(M)$

which implies that
$\Theta(k-1,D)+w(y_{t})+1=\operatorname{reg}(L)\leq\operatorname{reg}(M)$ by
the virtue of equation (6). Note that from assumption
$\operatorname{reg}(M\cap N)\leq\operatorname{reg}(M\oplus N)$, equations (7)
and (10) we obtain $\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})+1<\operatorname{reg}(M)$. Therefore using equation (8),
Lemma 4.1 and assumption
$\Theta(k,D\setminus\\{x_{t},y_{t}\\})<\Theta(k-1,D)+w(y_{t})+1$ we get that

$\Theta(k-1,D)+w(y_{t})+1=\operatorname{reg}(M)=\Theta(k,D\setminus\\{x_{t},y_{t}\\})+1.$

which implies

$\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})\\}<\Theta(k-1,D)+w(y_{t})+1.$

Subcase 2: Suppose $\operatorname{reg}(M\oplus N)<\operatorname{reg}(M\cap
N)$. Then applying regularity Lemma 2.8(6) on exact sequence (11) and using
equation (10) we get that

$\displaystyle\operatorname{reg}(J\cap
L)=\operatorname{reg}(M+N)=\operatorname{reg}(M\cap N)-1=\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})+1.$

Using $\Theta(k,D\setminus\\{x_{t},y_{t}\\})<\Theta(k-1,D)+w(y_{t})+1$,
$\operatorname{reg}(J\oplus L)=\operatorname{reg}(J\cap L)$, equations (5) and
(6), we have $\Theta(k-1,D)+w(y_{t})+1=\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})+1$ which implies
$\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})\\}<\Theta(k-1,D)+w(y_{t})+1.$
Thus Subcase 1  and Subcase 2  give that

$\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})\\}<\Theta(k-1,D)+w(y_{t})+1$

which further results in
$\operatorname{reg}(I(D)^{k})\leq\Theta(k-1,D)+w(y_{t})+1$, by the virtue of
equation (13). Now we will show that
$\operatorname{reg}(I(D)^{k})=\Theta(k-1,D)+w(y_{t})+1.$
Write, $I(D)^{k}=I_{1}+I_{2}$, where $I_{2}=(x_{t}y_{t}^{w(y_{t})})^{k}$,
$\mathcal{G}(I_{1})=\mathcal{G}(I(D)^{k})\setminus\mathcal{G}(I_{2})$ and
$I_{1}\cap I_{2}=I_{2}(I(D\setminus N_{D}[x_{t}])+(z))$. Further write,
$(I(D)^{k})^{\mathcal{P}}=(I_{1})^{\mathcal{P}}+(I_{2})^{\mathcal{P}}$, where
$(I(D)^{k})^{\mathcal{P}}$, $I_{1}^{\mathcal{P}}$ and $I_{2}^{\mathcal{P}}$
are the polarization of $I(D)^{k}$, $I_{1}$ and $I_{2}$ respectively. Thus
$(I(D)^{k})^{\mathcal{P}}=(I_{1})^{\mathcal{P}}+(I_{2})^{\mathcal{P}}$ is
Betti splitting by the virtue of [6, Corollary 2.7]. Then from [6, Corollary
2.2] we have

$\operatorname{reg}((I(D)^{\mathcal{P}}))=\operatorname{max}\\{\operatorname{reg}(I_{1}^{\mathcal{P}}),\operatorname{reg}(I_{2}^{\mathcal{P}}),\operatorname{reg}(I_{1}^{\mathcal{P}}\cap
I_{2}^{\mathcal{P}})-1\\}.$

Further,

$\displaystyle\operatorname{reg}(I_{1}^{\mathcal{P}}\cap
I_{2}^{\mathcal{P}})=$ $\displaystyle\operatorname{reg}((I_{1}\cap
I_{2})^{\mathcal{P}})$ $\displaystyle=$
$\displaystyle\operatorname{reg}(I_{1}\cap I_{2})\mbox{ (from Lemma
}\ref{lm5})$ $\displaystyle=$ $\displaystyle k(w(y_{t})+1)+\Theta(1,D\setminus
N_{D}[x_{t}])$ $\displaystyle=$
$\displaystyle(k-1)(w(y_{t})+1)+\Theta(1,D\setminus N_{D}[x_{t}])+w(y_{t})+1$
$\displaystyle=$ $\displaystyle\Theta(k-1,D)+w(y_{t})+2\mbox{ (from Lemma
}\ref{lm3.9}),$

equation (13), Lemma 2.4 and [9, Lemma 3.1 ] result in

$\operatorname{reg}(I_{1}^{\mathcal{P}})\leq\operatorname{reg}((I(D)^{k})^{\mathcal{P}})\leq\Theta(k-1,D)+w(y_{t})+1$

and Lemma 4.2 results in
$\operatorname{reg}(I_{2}^{\mathcal{P}})\leq\Theta(k-1,D)+w(y_{t})+1.$ Hence
using Lemma 2.4 we have
$\operatorname{reg}(I(D)^{k})=\Theta(k-1,D)+w(y_{t})+1.$ Thus from all the
cases, we obtain

$\Theta(k-1,D)+w(y_{t})+1\leq\operatorname{reg}(I(D^{k})).$

Now we will prove (3). Applying regularity Lemma 2.8(2) on exact sequence
(11), we get that

$\operatorname{reg}(M\cap
N)\leq\operatorname{max}\\{\operatorname{reg}(M\oplus
N),\operatorname{reg}(M+N)+1\\}.$

Thus from $\operatorname{reg}(M\cap N)=\operatorname{reg}(N)+1$ and
$\operatorname{reg}(M+N)=\operatorname{reg}(J\cap L)$ we have

$\displaystyle\operatorname{reg}(N)+1\leq\operatorname{max}\\{\operatorname{reg}(M),\operatorname{reg}(J\cap
L)+1\\}.$

Now applying regularity Lemma 2.8(2) on exact sequence (12) we get that

$\displaystyle\operatorname{reg}(N)+1$
$\displaystyle\leq\operatorname{max}\\{\operatorname{reg}(M),\operatorname{reg}(J\oplus
L)+1,\operatorname{reg}(I(D)^{k})+2\\}$

which implies that

$\displaystyle\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t})\leq\operatorname{max}\\{\Theta(k-1,D)+w(y_{t}),\operatorname{reg}(I(D)^{k})\\}$

by virtue of equations (5), (6), (7) and (9). Thus by (2), we get that

$\Theta(k,D\setminus N_{D}[x_{t}])+w(y_{t})\leq\operatorname{reg}(I(D)^{k}).$

Thus

$\displaystyle\Theta(k,D)=$
$\displaystyle\operatorname{max}\\{\Theta(k,D\setminus\\{x_{t},y_{t}\\}),\Theta(k,D\setminus
N_{D}[x_{t}])+w(y_{t}),\Theta(k-1,D)+w(y_{t})+1\\}$ $\displaystyle\leq$
$\displaystyle\operatorname{reg}(I(D^{k})).$

This proves the theorem. ∎

######  Corollary 4.4.

Let $D$ be an oriented unmixed forest. Let
$\\{\\{x_{1},y_{1}\\},\ldots,\\{x_{r},y_{r}\\}\\}$ be a perfect matching in
the underlying graph $G$ of $D$. Suppose $y_{1},\ldots,y_{r}$ are sinks. Then
for any $k\geq 1$,

$\displaystyle\operatorname{reg}(I(D)^{k})\leq(k-1)(w+1)+\operatorname{reg}(I(D))\mbox{
for any }k\geq 1,$

where $w=\operatorname{max}\\{w(x):x\in V(D)\\}$.

###### Proof.

Observe that $\Theta(k,D)\leq(k-1)(w+1)+\Theta(1,D)$, where
$w=\operatorname{max}\\{w(x):x\in V(D)\\}$ and from Theorem 3.6 we have
$\operatorname{reg}(I(D))=\Theta(1,D)$. Thus the Theorem 4.3 gives that

$\displaystyle\operatorname{reg}(I(D)^{k})=\Theta(k,D)\leq(k-1)(w+1)+\Theta(1,D)=(k-1)(w+1)+\operatorname{reg}(I(D)),$

for any $k\geq 1$. Thus

$\displaystyle\operatorname{reg}(I(D)^{k})\leq(k-1)(w+1)+\operatorname{reg}(I(D))\mbox{
for any }k\geq 1.$

∎

######  Remark 4.5.

Note that the formula for $\operatorname{reg}(I(D)^{k})$ in the Theorem 4.3 is
also equal to the following symmetric formula,

$\displaystyle\operatorname{max}\left\\{(w(y_{i_{l}})+1)k+\sum_{j=1,j\neq
l}^{s}w(y_{i_{j}})~{}:\text{ none of the edges
}\\{x_{i_{j}},y_{i_{j}}\\}\text{ are adjacent },y_{i_{j}}\in V(D)\right\\}$

for any $k\geq 1$.

######  Example 4.6.

Let $D$ be a Cohen-Macaulay weighted oriented graph with all leaves are sinks
as below.

Figure 2. Cohen-Macaulay weighted oriented forests

The weight of vertices are $w(x_{i})=1$, for all $1\leq i\leq 5$,
$w(y_{1})=3$, $w(y_{2})=4$, $w(y_{3})=3$, $w(y_{4})=3$, $w(y_{5})=2$. Then the
edge ideal of $D$ is

$I(D)=(x_{1}x_{2},x_{2}x_{3},x_{2}x_{4},x_{4}x_{5},x_{1}y_{1}^{3},x_{2}y_{2}^{4},x_{3}y_{3}^{3},x_{4}y_{4}^{3},x_{5}y_{5}^{2}).$

By the Theorem (4.3), for any $k\geq 1$,

$\begin{split}\operatorname{reg}(I(D)^{k})=&\operatorname{max}\Bigg{\\{}(\operatorname{max}\\{w(y_{1}),w(y_{3}),w(y_{4})\\}+1)(k-1)+w(y_{1})+w(y_{3})+w(y_{4})+1,\\\
&~{}~{}~{}(\operatorname{max}\\{w(y_{1}),w(y_{3}),w(y_{5})\\}+1)(k-1)+w(y_{1})+w(y_{3})+w(y_{5})+1,\\\
&~{}~{}~{}(\operatorname{max}\\{w(y_{2}),w(y_{5})\\}+1)(k-1)+w(y_{2})+w(y_{5})+1\\}\Bigg{\\}}\\\
=&\operatorname{max}\\{4(k-1)+10,~{}4(k-1)+9,~{}5(k-1)+7\\}\\\
=&\operatorname{max}\\{4(k-1)+10,~{}5(k-1)+7\\}.\end{split}$

This implies that $\operatorname{reg}(I(D)^{k})=4(k-1)+10$, for $1\leq k\leq
4$, and $\operatorname{reg}(I(D)^{k})=5(k-1)+10$, for all $k\geq 4$.

The formula in Theorem (4.3) does not hold for Cohen-Macaulay weighted
oriented forests with a leaf is not sink.

######  Example 4.7.

Let $D$ be a Cohen-Macaulay weighted oriented graph as given below.

Figure 3. Cohen-Macaulay weighted oriented forests

The weight of vertices are $w(x_{i})=1$, for $i=1,3,4$,
$w(x_{2})=7,w(x_{5})=6$, $w(y_{1})=5,w(y_{3})=3,w(y_{4})=4$ and $w(y_{i})=1$
for $i=2,5$.Then the edge ideal of $D$ is

$I(D)=(x_{1}x_{2}^{7},x_{2}^{7}x_{3},x_{4}x_{2}^{7},x_{4}y_{4}^{4},x_{1}y_{1}^{4},y_{2}x_{2}^{7},x_{3}y_{3}^{3},y_{5}x_{5}^{6},x_{4}y_{4}^{4}).$

Consider for any $k\geq 1$,

$\begin{split}\Theta(k,D)=&\operatorname{max}\Bigg{\\{}(\operatorname{max}\\{w(y_{1}),w(y_{3}),w(y_{4})\\}+1)(k-1)+w(y_{1})+w(y_{3})+w(y_{4})+1,\\\
&~{}~{}~{}(\operatorname{max}\\{w(y_{1}),w(y_{5})\\}+1)(k-1)+w(y_{1})+w(y_{5})+1,\\\
&~{}~{}~{}(\operatorname{max}\\{w(y_{2}),w(y_{5})\\}+1)(k-1)+w(y_{2})+w(y_{5})+1\\}\\\
&~{}~{}~{}(\operatorname{max}\\{w(y_{4}),w(y_{5})\\}+1)(k-1)+w(y_{4})+w(y_{5})+1\\}\Bigg{\\}}\\\
&=\operatorname{max}\\{6(k-1)+11,~{}6(k-1)+7,~{}2(k-1)+3,~{}5(k-1)+6\\}\\\
&=6(k-1)+11.\end{split}$

This implies that $\Theta(k,D)=6(k-1)+11$, for $k\geq 1$. Now, using Macaulay
$2$ [8] we have $\operatorname{reg}(I(D))=24\neq\Theta(1,D)=11$,
$\operatorname{reg}(I(D)^{2})=31\neq\Theta(2,D)=17$.

Acknowledgement: Manohar Kumar is thankful to the Government of India for
supporting him in this work through the Prime Minister Research Fellowship.
Data availability statement Data sharing is not applicable to this article as
no datasets were generated or analyzed during the current study.

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# Concerning the Verity of the MMRD Relation for Novae

Allen W. Shafter Department of Astronomy, San Diego State University, San
Diego, CA 92182, USA  J. Grace Clark Department of Astronomy, San Diego State
University, San Diego, CA 92182, USA  Kamil Hornoch Astronomical Institute of
the Czech Academy of Sciences, Fričova 298, CZ-251 65 Ondřejov, Czech Republic

###### Abstract

It has long been claimed that novae reaching the highest luminosity at the
peak of their eruptions appear to fade the fastest from maximum light. The
relationship between peak brightness and fade rate is known as the Maximum-
Magnitude, Rate-of-Decline (MMRD) relation. Lightcurve parameters for the most
recent sample of M31 recurrent novae are presented and used to buttress the
case that the observed MMRD relation can be explained as a consequence of
observational selection effects coupled with expectations from standard nova
models.

Cataclysmic Variable Stars (203) – Novae (1127) – Recurrent Novae (1366)

##  1 Introduction

Mclaughlin (1945) was first to argue that the peak luminosity of a classical
nova was correlated with its rate of decline from maximum light. Over the
years, the correlation has come to be known as the Maximum-Magnitude versus
Rate-of-Decline (MMRD) relation, and has been calibrated many times, both in
the Galaxy and in M31 (e.g., Cohen, 1985; Capaccioli et al., 1989; Downes &
Duerbeck, 2000; Shafter et al., 2011; Özdönmez et al., 2018). Despite its long
history, the verity of the MMRD has been called into question in recent years.

In a sample of M31 novae, Kasliwal et al. (2011) found several systems that
were fainter for a given decline rate than predicted by the MMRD. They
considered the possibility that these objects might be recurrent novae (RNe),
but noted that most had spectroscopic types (Fe II) that were inconsistent
with that interpretation.

Figure 1: The MMRD plane divided into four quadrants: Q1 – Q4. The filled
black circles show data for Galactic novae from Downes & Duerbeck (2000). The
dashed line is the best linear fit to these data. The blue squares show
Galactic RNe from Schaefer (2010), while the red diamonds show our updated M31
RN sample. Most classical novae discovered in routine nova patrols fall either
in Q1 or Q4. Known RNe on the other hand fall almost exclusively in Q2, while
Q3 is almost devoid of novae.

Subsequent to the Kasliwal et al. (2011) study, Özdönmez et al. (2018)
compiled an extensive database for Galactic novae showing that the MMRD
relation was generally followed and remained “a useful tool for statistical
analyses”. Shortly thereafter, Schaefer (2018) recalibrated the Galactic MMRD
based on Gaia distances concluding that the MMRD relation was plagued by
considerable scatter with a fit too poor to be usable for distance
determinations.

In this note, we argue that the MMRD relation can be best understood as a
combination of observational selection effects coupled with standard
predictions from nova theory.

##  2 The Nature of the MMRD

To understand the placement of novae in the $M$ (max) – log $t_{2}$ plane
(hereafter the MMRD plane) we first consider how the ignition mass – the
accreted mass required to trigger a thermonuclear runaway (TNR) on the white
dwarf – depends on properties of the progenitor binary. To first order,
$M_{\mathrm{ign}}$ depends only on the pressure at the base of the accreted
layer, which is a function of the WD mass. However, the temperature of the
accreted layer is also important, and it is strongly affected by the rate of
accretion onto the white dwarf. Thus, as first considered by Nomoto (1982) and
explored by many groups since (e.g., Townsley & Bildsten, 2005; Wolf et al.,
2013; Kato et al., 2014), the mass necessary to trigger a TNR depends on both
the WD mass and the rate of accretion onto its surface.

For illustrative purposes, we consider in Figure 1 the MMRD relation for the
sample of Galactic novae studied by Downes & Duerbeck (2000). In addition, we
show separately the known Galactic RNe from Schaefer (2010), along with the
most recent M31 RN sample presented here together for the first time. Despite
some scatter, the Downes & Duerbeck nova sample follows the best-fitting MMRD
relation (dashed line) quite well. However, the RNe fall consistently below
the MMRD relation.

To explore the observed properties of the MMRD in detail, it is useful to
divide the MMRD plane into four quadrants: (Q1) an upper left quadrant
consisting of relatively fast and bright novae, (Q2) a lower left quadrant
that includes fast and faint novae, (Q3) an upper right quadrant where slowly
evolving luminous novae should lie, and finally (Q4) a lower right quadrant
where slowly evolving and faint novae are found.

Given a population of nova progenitors with a range of WD masses and accretion
rates, one can imagine systems occupying all quadrants of the MMRD plane.
However, we argue below that the observed MMRD relation arises because two of
the quadrants, the second and especially the third, are selected against.

In the case of the second quadrant, the low luminosity and fast evolution
(short $t_{2}$) suggests a weak TNR and a relatively small ejected (and
ignition) mass. Models show that novae with these properties arise from
systems with high mass white dwarfs accreting at high rates. The small
ignition masses and high accretion rates produce novae with the shortest
recurrence times (e.g., see Kato et al., 2014, their figure 6). Thus, it is
not surprising that the known RNe are found in the lower left quadrant of the
MMRD plane. Considering their short recurrence times, it is reasonable to
wonder why this quadrant of the MMRD plane is not more heavily populated. The
answer lies in the fact that faint and fast novae are strongly selected
against in typical nova surveys, most of which have relatively bright limiting
magnitudes and coarse temporal coverage.

In the third quadrant of the MMRD plane we expect to find novae that are
luminous and slowly evolving. The slow evolution suggests a massive ejecta
(and ignition mass), while the high luminosity implies a strong TNR. Models
suggest that the progenitors of such novae contain relatively low mass white
dwarfs accreting at low rates. The slow accumulation of matter on the white
dwarf and a high ignition mass results in both a strong TNR and a very long
recurrence time. Such systems are the polar opposite of the RNe. Although they
should have bright eruptions, the recurrence times are expected to be
exceedingly long. Thus, systems in the upper right quadrant of the MMRD are
expected to erupt extremely rarely, in agreement with observation.

Consistent with the nature of the MMRD relation, most novae fall into either
the first or the fourth quadrants. Novae in the first quadrant presumably
arise from novae with high mass white dwarfs accreting at relatively low
rates, while novae in the fourth quadrant likely have progenitors that contain
relatively low mass white dwarfs accreting at relatively high rates. In the
first case, the high mass WD and the slow accretion will result in an accreted
layer that is highly degenerate at the time when the TNR ensues. Such novae
should appear relatively bright and evolve quickly. Conversely, the rapidly
accreting low mass WD systems will be characterized by less degenerate
accreted layers and a weaker TNR resulting in less luminous novae with a
generally slower evolution.

In summary, the MMRD emerges as a result of observational selection against
faint and fast (recurrent) novae coupled with a dearth of eruptions from
systems with extremely long recurrence times.

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Table A1: MMRD Lightcurve Parameters (Figure 1) Nova |  $t_{2}$ (d) |  $M$(max)aa$M$(max) based on $m-M=24.42$ (Tully et al., 2013) and $A_{R}=0.14$ (Schlafly & Finkbeiner, 2011) for M31. | Filter | Notebb(1) Downes & Duerbeck (2000); (2) Schaefer (2010); (3) This work.
---|---|---|---|---
Galactic Nova Sample
V500 Aql | 17.0 | $-8.70\pm 0.40$ | pg | 1
V603 Aql | 4.0 | $-8.90\pm 0.20$ | $V$ | 1
V1229 Aql | 20.0 | $-6.70\pm 0.95$ | $V$ | 1
T Aur | 45.0 | $-6.80\pm 0.50$ | pg | 1
V842 Cen | 35.0 | $-7.40\pm 1.00$ | $V$ | 1
V450 Cyg | 88.0 | $-6.50\pm 0.60$ | $V$ | 1
V476 Cyg | 6.0 | $-9.90\pm 0.60$ | $V$ | 1
V1500 Vyg | 2.4 | $-10.70\pm 0.50$ | $V$ | 1
V1819 Cyg | 37.0 | $-6.80\pm 0.50$ | $V$ | 1
V1974 Cyg | 17.0 | $-8.00\pm 0.10$ | $V$ | 1
HR Del | 172.0 | $-6.10\pm 0.40$ | $V$ | 1
DQ Her | 39.0 | $-7.50\pm 0.30$ | $V$ | 1
V446 Her | 7.0 | $-9.90\pm 0.20$ | pg | 1
V533 Her | 22.0 | $-7.50\pm 0.50$ | $V$ | 1
CP Lac | 5.3 | $-9.30$ | $V$ | 1
DK Lac | 11.0 | $-9.60\pm 0.50$ | $V$ | 1
GK Per | 7.0 | $-9.00\pm 0.30$ | $V$ | 1
RR Pic | 20.0 | $-7.80\pm 0.30$ | $V$ | 1
CP Pup | 6.0 | $-10.70\pm 0.50$ | $V$ | 1
V351 Pup | 10.0 | $-8.40\pm 0.50$ | $V$ | 1
FH Ser | 42.0 | $-7.30\pm 0.50$ | $V$ | 1
XX Tau | 24.0 | $-8.40\pm 0.75$ | pg | 1
RW UMi | 48.0 | $-7.80\pm 0.90$ | pg | 1
LV Vul | 21.0 | $-7.00\pm 0.45$ | $V$ | 1
NQ Vul | 23.0 | $-6.10\pm 0.80$ | $V$ | 1
PW Vul | 82.0 | $-6.70\pm 0.40$ | $V$ | 1
QU Vul | 22.0 | $-7.50\pm 0.20$ | $V$ | 1
QV Vul | 50.0 | $-6.40$ | $V$ | 1
Galactic Recurrent Novae
T Pyx | 32.0 | $-7.10$ | $V$ | 2
IM Nor | 50.0 | $-6.60$ | $V$ | 2
CI Aql | 25.4 | $-7.10$ | $V$ | 2
V2487 Oph | 6.2 | $-7.40$ | $V$ | 2
U Sco | 1.2 | $-8.50$ | $V$ | 2
V394 CrA | 2.4 | $-8.40$ | $V$ | 2
T CrB | 4.0 | $-7.30$ | $V$ | 2
RS Oph | 6.8 | $-10.60$ | $V$ | 2
V745 Sco | 6.2 | $-8.00$ | $V$ | 2
V3890 Sgr | 6.4 | $-8.60$ | $V$ | 2
M31 Recurrent Novae
M31N 1923-12c | $12.6\pm 5.8$ | $-6.96\pm 0.11$ | $R$ | 3
M31N 1926-07c | $11.1\pm 1.5$ | $-7.01\pm 0.16$ | $R$ | 3
M31N 1960-12a | $4.7\pm 0.6$ | $-6.76\pm 0.11$ | $R$ | 3
M31N 1963-09c | $3.9\pm 0.6$ | $-6.61\pm 0.11$ | $R$ | 3
M31N 1966-09e | $81.0\pm 11.0$ | $-6.44\pm 0.11$ | $R$ | 3
M31N 1984-07a | $10.7\pm 1.3$ | $-8.35\pm 0.10$ | $R$ | 3
M31N 1990-10a | $11.1\pm 1.7$ | $-7.00\pm 0.11$ | $R$ | 3
M31N 1997-11k | $88.0\pm 9.0$ | $-5.99\pm 0.20$ | $R$ | 3
M31N 2006-11c | $1.8\pm 0.5$ | $-8.02\pm 0.11$ | $R$ | 3
M31N 2007-10b | $2.3\pm 0.5$ | $-7.75\pm 0.11$ | $R$ | 3
M31N 2007-11f | $10.2\pm 2.6$ | $-7.22\pm 0.11$ | $R$ | 3
M31N 2008-12a | $2.2\pm 0.1$ | $-6.25\pm 0.04$ | $R$ | 3
M31N 2017-01e | $6.0\pm 0.6$ | $-6.76\pm 0.11$ | $R$ | 3