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CHAPTER I

INTRODUCTION

1.1BackgroundAllah sent down the Qur'an with its multiple functions. Among these

functions is al-Quran as a guide (hudan), distinguishing between right and

wrong (al-furqan), healing heart disease (syifa), advice (mau'izhah), and

sources of information (bayan). As a source of information, the Qur'an teaches

many things to humans ranging from about beliefs, morals, principles of

worship, to the principles of science, which includes science and math.

One word of Allah that gives motivation to learn science and math is

like include in the letter listed the following five verses of Yunus:

FSWA[[",=lW=TWU,[n4Sr;TWUjrWP$;Q#(P,Sj[k[D


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studies on how to calculate or measure something with numbers, symbols, or

number.

Mathematics is very influential in the development of other sciences. For

example, it is influence in physics, biology and other sciences. Expert from

different disciplines use mathematics to various needs related to their scholarship.

The universe contains the forms and concepts of mathematics, although

mathematics of the universe is created before it existed. The universe and all its

contents were created by God with careful measurements and calculations

accurately with a steady, and with formulas and equations are balanced and neat

(Abdussyakir, 2007: 79).

In Al-Qamar epistle stated:

4"Z\[P;QOUj[r

Meaning:

That is "We created everything by size"

God created the universe and its contents with the size of a careful and

thorough. God creates the air movement has a measure of speed, how much and

where a speed direction. If the quantities in a natural process related to each other,

then they are relationship can be formulated in mathematical form.

In another verse stated:

OU$0WS[,r(]WT1UWTlc)d;jUWT

0UWTBddos ,P[\WT"ZPWrjUU>ndjU!


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Meaning:

"What belongs to Him is the kingdom of heaven and earth, and he did not have

children, and no ally for him in the power of (His), and he has created everything,

and he establishes the measures with a neat" (Al-Furqan: 2).

The above verse explains that everything in nature there is found a

measurement, there is found a count, there is found a formula, or there is found an

equation. Mathematician or a physicist does not create a formula or equation. The

formulas which found are not created by people themselves, but have provided.

Humans only discovered and symbolize in the language of mathematics

(Abdussyakir, 1997: 80).

Graph theory is a branch of mathematics commonly used in everyday life.

For example, it is used in the manufacture of transport routes of travel, the

schedule settings, grid settings, and so forth. Concept and results of the

corresponding graph theory, introduced with the intention of making a

mathematical simulation problem. This is what makes the interesting graph theory

and the more developed by the mathematicians.

Labeling of graphs is a topic in graph theory. Object of study in the formof graphs that are generally represented by dots and side and a subset of the

original number called a label. First introduced by Sadlk (1964), and Stewart

(1966), Kotzig and Rosa (1970). Until now the use of labeling theory of graphs is

perceived role, especially in the sectors of communication and transportation

systems, geographic navigation, radar, computer data storage, integrated circuit

design and electronic components.

The uniqueness of graph theory is the simplicity of the subject studied,

because it can be shown as dots (vertices) and lines (edges). Although the subject

of graph theory topics is very simple, but the content within it is has not been

simple. Complexities and problems are always there and even today there are still

many unsolved problems (R. Gunawan S, 2002:1)


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Many topics of discussion of unsolved makes scientists want to use the

theories or theorems about the discussion have become obstructed. Because it is

impossible to use a science that has not been clearly truth. As described in the

word of God, especially in verse 36 of Al-Isra epistle which reads:

YWTU!$_kU[UO1D[,Wo^WT

[kZUWTr"[UTDZO;


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1V

nV

5V

4V

6V

3V

2V

In previous research, it has been found a consecutive labeling on

graph. The authors will be develops a super edge magic labeling of graphs

which are connected with the hook point. From some of the discussion above, the

writer interest in writing about the "Super Edge Magic Labeling on graph that

is connected with the hook vertex."

1.2 Problem Formulation

The formulation of the problem to be discussed in this paper is: what about

super edge magic labeling on graph that is connected with the hook vertex?

1.3Research Objective

Based on the formulation of the problem above, the purpose of this thesis

is to analyze the super edge magic labeling on graphthat is connected with the

hook vertex.

1.4Limitation Problem

In order to the discussion in this paper does not extend, the authors in this

research is restrict the object of study in super edge magic labeling of graph for

2 to 4 that are connected with the hook vertex like this:


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1.5 Benefits of Research

The result is expected to be useful to:

1. For AuthorsThis research is an opportunity for authors to add information and broaden

the knowledge of accepted theories in college, especially about graphs theory.2. For Readers

As a material to increase the knowledge of mathematics, especially about

graph theory, and is expected to be a reference for future research.

3. For InstitutionsAs an addition the literature about the graph theory and as an addition

reference for lecture material.

1.6 Research MethodsThe steps to be undertaken in this research are:

1. Formulate problemBefore the author brings about this research, the author will arrange the plan of

the research of a problem about the super edge magic labeling on graph.

2. Collecting data

Researchers collected data in the form of primary data and secondary data.

Primary data in this study were obtained from direct observations which author

doing, there are some of images of graph, many edges, many vertices, and super

edge magic labeling on star graph. The secondary data that used in this research


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are the definitions, theorems, properties of graphs, etc., from some of the literature

including books, documents, thesis before, and others.

3. Analyzing data

The steps taken to analyze the data in this paper are:

1. Trying to do a super edge magic labeling on graph that is connected with

the hook vertex.

2. Observe the patterns of super edge magic labeling on graph that is

connected with the hook vertex. This pattern obtained is still considered a

conjecture.

3. The conjectures that obtained stated in the mathematically sentence that is

proved.

4. Give final conclusions from the research.

1.7 Systematic Writing

Writing this paper will be divided into several chapters. The division

compositions of the chapters are:

CHAPTER I: Introduction. This chapter discusses the background, problem

formulation, research objectives, limitation problems, the benefits of research,

research methods, and systematic of writing.

CHAPTER II: Basic Theory. This chapter contains the basics of the theory to be

used in subsequent chapters, such as the definition of graph, functions, star graph,

super edge magic labeling, as well as other theories that help.

CHAPTER III: Discussion. This chapter discusses about the super edge magic

labeling on graph that is connected with the hook vertex.

CHAPTER IV: Conclusion. This chapter contains the conclusions of the

materials that have been discussed in previous chapters.


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CHAPTER II

BASIC THEORY

2.1GraphDefinition 2.1:

A graph is a diagram consisting of points, called vertices, joined together

by lines, called edge; each edge joins exactly two vertices (Watkins and Wilson,

1990: 8).

Definition 2.2:

A graph is a finite nonempty set of objects called vertices (the singular

is vertex) together with a (possibly empty) set of unordered pairs of distinct

vertices of called edges. The vertex set of G is denoted by , while the edge

set is denoted by (Chartrand and Lesniak, 1986: 4).

Definition 2.3:

A graph consists of a non-empty set of elements, called vertices, and a

list of unordered pairs of these elements, called edges. The set of vertices of the

graph is called the vertex-set of, denoted by and the list of edges is

called the edge list of, denoted by Ifand are vertices of , then an

edge of the form oris said to join and (Watkins and Wilson, 1990: 8).


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Definition 2.4:

A graph is an ordered pair in which is a set called the vertex set

and is a set consisting of subsets of with two elements. The set is called the

edge set, and we say that and are joined by an edge if belongs to

(Witala, Stephen, 1987: 178) .

Example:

Consider the graph where and

There graph can be represented in any of the three

ways shown in figure:

Picture 2.1 graph

Although the diagram look different, we should note that they all have the

same properties; in fact, if the bottom two were straightened out, the result in each

cases would be the diagram on top. In each diagram it is possible to go from to

by means of a route which takes us through and ; and if any of the segments

are broken, it will no longer be possible to go from to .

The Qur'an is the source of all knowledge. No one can deny that in the

Qur'an not only put the basics of human life in the regulations to do with God the

Creator, in its interaction with fellow human beings and in their action of nature

around him, but also stated to what man was created.


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The vertex in a graph can be assumed according to need in solving a

problem. If two points on a graph is assumed as an object and is connected to a

edge, then this has the sense that these two objects have a relationship. The

definition of a graph can be represented in human relations and the genie with the

Creator. In verse 56 of Adh-Dzariat we find the statement of Allah SWT as

follows:

$WT/P[\BI_4/_WT

YDTjWl

The meanings: "And I did not create the genie and men but that they may serve

Me."

The meaning of "worship" here is to lose yourself, not just prayer, but

doing all that he commanded and the like, including everything that banning

abstinence and did not like, at least as befits a 'servant' or servants behave towards

their owners (Ahmad Baiquni. 1995: 66).

The above verse can be represented in the definition of graph, with

illustrated as follows:

Figure 2.2 the relationship between God and his servant

2.2Adjacent and incidentDefinition 2.5

The edge is said to join the vertices and . If is

and edge of graph, then and are adjacent vertices, while and are


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incident, as are and . Furthermore, if and are distinct edges of incident

with a common vertex, than and are adjacent edges. It is convenient to

henceforth denote an edge by or rather than by (Abdussakir, dkk.

2009: 6).

e 2e Figure 2.3 adjacent and incident

Description:

and , and are adjacent

and are incident with

and are incident with

and are adjacent

The cardinality of the vertex set of a graph is called the order ofand is

denoted by , or more simply, , while the cardinality of its edge set is the

size of and is denoted by or. A graph has order and size . The

degree of a vertex is the number of edges to which it is incident.

According to the Islamic context, adjacent and incident can be represented

in worship Sa'i. In Al-Baqarah verse 158 Allah says:

DZPQWTn[,WTB$

n[ZB[,UM[O

^/kTTn[,(ZU

[[Q;AOlPDTsSd

[,IB$WTSUU!


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willingness the heart, And Allah is verily grateful for kindness and also

a Knower. "

Literally meaning of Sai is a business, while the syariah meaning on hajj

and Umrah is back and forth seven times between Mount Safa and Marwah for the

sake of carrying out the commandments of God (Shihab, 2000:345).

Implementation of Sa'i between Safa and Marwah can be represented in the

adjacent. This can be illustrated with represent two hills Safa and Marwah, which

are as two adjacent vertexs (represented by sa'i).

Figure 2.4 Representation adjacent to the Process of Worship

2.3 The Examples of Common Graphs:

1. Complete GraphsA complete graph is a graph in which every two distinct vertices are joined

by exactly one edge. The complete graph with n vertices is denoted by .

K2

K3

K4

K Figure 2.5 complete graph

2. Null Graphs

A null graph is a graph containing no edges. The null graph with n vertices

is denoted by .


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1N N 3N 4N

Figure .6 null graph

3. Cycle Graphs

A cycle graph is a graph consisting of a single cycle. The cycle graph with

n vertices is denoted by .

1C

2C 3C 4C

Figure 2.7 cycle graph

4. Path Graphs

A path graph is a graph consisting of a single path. The path graph with n

vertices is denoted by . Note that has n-1 edges, and can be obtained from

the cycle graph by removing any edge.

P3

P 4P1P

Figure 2.8 path graph

5. Bipartite Graph

A bipartite graph is a graph whose vertex-set can be split into sets A and

Bin such a way that each edge of the graph joins a vertex in A to a vertex in B.

We can distinguish the vertices in A from those in B by drawing the former in


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black and the latter in white, so that each edge is incident with a black vertex and

white vertex. The example of bipartite graph is:

Figure 2.9 bipartite graph

A complete bipartite graph is a bipartite graph in which each black vertex

is joined to each white vertex by exactly one edge. The complete bipartite graph

with r black vertices and s white vertices is denoted by . A complete bipartite

graph of the form of is called a star graph. Some examples of complete

bipartite graphs are:

K 2,2K

Figure 2.10 complete bipartite graph

Complete graph illustrating can be taken from a verse which describes the

relationship among humans. Word of God in Q.S Al Hujuraat verse 13:

RMjTd4WT


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0Q;[[AWT>S

"UWTSWr[)D

$nT[j;0U!TD.on[\

It means: O man, Indeed we created you from a male and a female and made you

a nation-peoples and tribes that you may know each other. Indeed, the

most honorable among you is the most god-fearing among you. Indeed

Allah is a Knowing.

This can be represented in the form of a graph with the tribes or nations as

a vertex. For example take kinds tribes or nations, then it has vertex. The

shape relationship to "know each other" is considered as a edge connecting each

tribes or nations. Because as explained in the letter of Al-Hujuraat verse 13 that

people should know each other, so the between a vertex with another vertex which

must also be mutually connected. So if the connection between the tribes was

described, will be in to the picture as follows:

Figure 2.11 the representation of a complete graph on human relations

2.4 Functions

Definition 2.6

Let and be two non empty sets. A function from into , denoted

by , is a rule that assigns to every element in a unique element in .

The set is the domain of the function and the set is its codomain. If y is the

unique element in assigned by the function to the element , we say that y is


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the image of and is preimage of y and we write . The set is

called the range of the function. The range of function is a subset of its codomain

(Balakrishnan, 1991: 7).

Example 2.1:

The set is a function from to

. Each element of is assigned the unique value ; 2 is assigned to

unique value , and 3 is assigned the unique value . We can depict the situation

as shown in figure 2.2.1, where an arrow from means that we assign the

letter to the integer. We call a picture such as figure 2.2.1 an arrow diagram.

f

X Y

Figure 2.12 The arrow diagram of the function of example 2.1

Example 2.2:

The set is not a function from to

because the element 4 in is not assigned to an element in . It is

also apparent from the arrow diagram (see the figure 2.2) that this set is not a

function because there is no arrow from 4.

f

X Y


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Figure 2.13 The arrow diagram of the set in example 2.2

Example 2.3:

Let be the function defined by the rule

For example,

Although we frequently find functions defined in this way, the definition is

incomplete since the domain and codomain are not specified. If we are told that

the domain is the set of all real numbers and the codomain is the set of all

nonnegative real numbers, in ordered-pair notation, we would have

The range of is the set of all nonnegative real numbers.

Definition 2.7

A function from to is said to be one-to-one (or injektif) if for each ,

there is at most one with

Example 2.4:

The function

From to is one to one. If a function from to is

one to one, each element in in its arrow diagram will have at most one arrow

pointing to it (see figure 2.14)


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f

X Y Figure 2.14 The function of example 2.4

Example 2.5:

The function

Is not one-to-one since . If a function is not one-to-one, some

element in in its arrow diagram will have two or more arrows pointing to it (see

figure 2.15)

f

X Y Figure 2.15 A function that is not-one-to-one

Definition 2.8

If a function from to and the range of is , is said to be onto (or an onto

function or a surjective function).

Example 2.6

The function


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From to is one-to-one and onto .

f

X

Figure 2.16 A function that is one-to-one and onto

Example 2.7

The function

From to is not onto .

f

X Y Figure 2.18 A function that is not onto

Definition 2.2.4

A function that is both one-to-one and onto is called a bijection.

Example 2.8

The function of example 2.6 is a bijection.


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Example 2.9

If is a bijection from a finite set to a finite set , then , that is the

sets have the same cardinality and the are the same size. For example,

Is a bijection from to . Both sets have four elements.

In effect, counts the elements in is the first element in

is the second element in ; and so on.

2.5LabelingGiven a graph and . A labeling for a graph is a

map that takes graph elements to numbers (usually positive or non-negative

integers). Kotzig and Rosa defined a magic labeling to be a total labeling on the

vertices and edges in which the labels are the integers from 1 to

The sum of labels on an edge and its two endpoints is constant. In 1996 Ringel

and Llado redefined this type of labeling as edge-magic. Also, Enomoto etalhave

introduced the name super edge-magic for magic labelings with the added

property that the vertices receive the smaller labels,

A one-to-one map from onto the integers is an

edge-magic labeling if there is a constant so that for any edge there are

The constantis called the edge magic number

for An edge-magic labeling is called super edge-magic if

and A graph is called edge-

magic (resp. super edge-magic) if there exists an edge-magic (resp. super edge-

magic) labeling of.

Consider a graph and . The order of is , and

the size of is . A total edge magic labeling on graph is a bijection function

from onto the integers


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So that for any edge

for any constant k. The constant is called the edge magic number for and

called total edge magic. (Wijaya and Baskoro, 2000:159 and Miller, 2000: 168).

For example, consider a graph , and

. So, the order ofis , and the size of is . Will be

shown that graph is a total edge magic labeling.

Picture 2.18 graph

If created a function from onto set , can be drawn

as follows:

f

Picture 2.19 the function from to set

Obtained:


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So the function is a total edge magic labeling in. Labeling the graph in

order to obtain a total edge magic labeling can be drawn as follow:

Picture 2.20 total edge magic labeling

Magic total labeling, which maps the vertex set of a graph into the set

is called a super edge magic labeling. Thus, super edge

magic labeling is a special form of total edge magic labeling. Each super edge

magic labeling is certainly total edge magic labeling, but not on the contrary. The

Graph that can be subject to super edge magic labeling is called super edge magic

labeling graph (Abdussakir, 2005: 27).

For example, consider the figures bellow:

Figure 2.21 figure 2.22

Figure 2.21 and figure 2.22 are the total edge magic labeling.

Nevertheless, the labeling on the figure 2.21 is called super edge magic labeling,

while in figure 2.22 is not a super edge magic labeling. This is because the set of

point on figure 1 is mapped to the set while not like this about figure

2.22.


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Worship fardhu praying is an example of one of worship that can be

represented in graph labeling. Prayer has a very important position in Islam, and is

the foundation for the establishment of religion Islam. Worship Prayer in Islam is

very important, so that should be implemented in time.

Al-Quran does not explain in detail the times the fardhu praying.

However, this times has been revealed in detail on the hadith of Prophet

Muhammad, initial limit until the deadline of each times. One of the hadiths that

explain the times of fardhu praying is the hadith narrated by Ahmad Nasa'i and

At-Turmudzi from Jabir ibn Abdullah ra are as follows:

"Behold, Gabriel came to Prophet Muhammad, and said to him:" get up

and take a pray" then the Prophet take a dzuhur pray when sun has slipped. Then

Gabriel also came to the Prophet at the time 'Asr, then said: "wake up and take a

pray ". So the prophet take a pray when shadow of everything was all him. Then

Gabriel came to Prophet also at maghrib time, then said: "wake up and take a pray

" then the prophet take a maghrib pray at a time when the sun had set. Then

Gabriel came in the evening time, then said: "wake up and take a pray " then the

prophet taken isya pray at a time when already lost the red clouds. Then Gabriel

came also at the time of shubuh, when the sun been brilliant dawn. The next day

Gabriel came again to pray of dhuhur. He said: "wake up and take a pray " then

the prophet taken dzuhur pray when the shadow of all things has become all

himself. Then Gabriel came again at the time of Asr, then said: "wake up and take

a pray " then the prophet take a Asr pray when the shadow of all things has

happened twice from her shadow. Then Gabriel came again at the maghrib same

time as the time he arrived yesterday. Then Gabriel came again in the evening

time when has passed half the night, or third night, then the prophet take a isya

pray. Then Gabriel came again at dawn was shining brightly, and said: "wake up

and take a pray ", then the prophet woke up and take a shubuh praying. After that

Gabriel said: "the times in between these two this time, it is time to pray" (book of

hadith Imam Ahmad, hadith to 10 819).


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Based on the above hadith, it can be specified a provisions of fardhu

prayer time. For dzuhur time, starting from the moment the sun slipped after

reaching a culmination point in the daily circulation until the shadow of

something the same length. Asr time begins from the shadow of something similar

in length to sunset. Maghrib time begins from sunset until the disappearance of

red cloud. Evening (isya) time starts since the loss of red cloud until half the

night (dawn), and the Fajr (shubuh) time starts from dawn sunrise to sunrise. So if

the times of fardhu praying labeling is represented in the graph are drawn as

follows:

Figure 2.23 representing graph labeling on times of fardhu praying

2.4 Star Graph

Star graph ( is complete bipartite graph which is formed by graph.

Here is an example of graph:

V

nV5

V

4V

6V

3V

2V

Figure 2.24 graph


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A star graph depicted on a related word of God sunatullah. In a

epistle of An-Nahl verse 11 says that:

/


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DAFTAR PUSTAKA

Abdussakir, dkk. 2009. Teorigraf.Malang: UIN Malang Perss

Abdussyakir. 2007. Ketika Kyai Mengajar Matematika. Malang: UIN Malang

Press.

Baiquni, Ahmad. 1995. Al-Quranilmu PengetahuandanTeknologi. Yogyakarta:

PT Dana Bhakti Prima Yasa.

Balakrishnan, V. K. 1991. Introductory Discrete Mathematics. New Jersey:

Prentice-Hall, Inc.

Chartrand, G. and Lesniak, L.1986. Graph and Digraph second Edition.

California: Wadsworth. Inc

Park, Ji Yeon; Choi, Jin Hyuk, and Bae, Jae-hyeong. 2008. On SuperEdge Magic

labeling Of Some Graphs.

(http://icms.kaist.ac.kr/mathnet/thesis_file/02_B05-1206.pdf. diakses 27

agustus 2010)

Ross, Kenneth A. and Wright, Charles R.B. 1992. Discrete MathematicsThird

Edition. New Jersey: Prentice Hall International, Inc.

Wilson, R.J. and Watkins, J. J. 1990. GraphsAnIntroductoryApproach. Canada:

John Wiley and Sons, Inc.

Wiitala, Stephen A. 1987. Discrete MathematicsA UnifiedApproach. Singapore:

Mc. Graw-Hill, Inc.


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