Journey In Graph Theory (Too Shi Wei, 278803)

WHAT WHO
ABOUT ARE
THEM? THEM?

WHEN
SHE
DIE?

HOW IS LECTURER:
THE PROF. DR.
HASLINDA BT
MURDER? IBRAHIM

TOO SHI WEI 278803

CONTENT

INTRODUCTION 1
2
WHO ARE THEM?
10
- EXPLANATION 17
- HANDSHAKING THEOREM 30
- ADJACENCY MATIRX
- INCIDENCE MATRIX

WHAT ABOUT THEM?

- SPANNING TREE

WHEN SHE DIED?

- TIMELINE
- GRAPH COLORING

HOW IS THE MURDER?

- SHORTEST PATH

INTRODUCTION

“Hello 911, I would like to report that someone died at Deluna
Homestay!”

On 4/6/2020, about 11 pm the officer received a call that reported
someone died. When he arrives at the scene, the dead are killed by
excessive bleeding, with a knife in her heart. The dead named Moon is a
popular star who received a lot of goals and had a very impressive
performance. In Deluna Homestay, there are 6 persons which are 5
suspects and the dead only.

Today is Moon’s Birthday. She invites her friends, workmate and
manager to join her birthday party. Lucas, Aria and Zoey are Moon’s
friends while Robin is her workmate. Stella is Moon’s agent which has
the responsibility to take care of and guide Moon’s daily life. They are
invited and arrive at Deluna Hotel to celebrate Moon’s Birthday and
will stay at the homestay for a night. Before celebrating her birthday,
Moon died in her room.

Who will be the murderer? Why did he or she want to kill her? How did
the murderer kill her?

There are many questions for the officer so he decided to use the
Mathematics Method to solve this murder case.

1

WHO ARE
THEM?

2

EXPLANATION

The graph G=(V,E) consists of V which
represents the set of vertices while E
represents the set of edges. This is
multigraph, it consists of undirected edges
and multiple edges.

Through the graph, the character of the
vertices, V and the edges, E is the
relationship between the vertices.

Moon as the vertices a because she is the
main character in this murderer case. She
knows 5 other people, who are her friends,
workmate and her agent.

Then Aria is the wife of Lucas. They are a

couple. For Zoey with Aria and Moon also,

they are besties during school time. Next,

Robin knows Lucas and Aria since he always

heard the Moon mentioned about them.

Lastly, Stella is the agent for Moon that in

charge of her daily life. 3

To sort out the relationship more clearly, the
officer draws a graph by using a, b, c, d, e
and f to represent the dead and suspects.

a= Moon
b= Lucas
c=Aria
d=Zoey
e=Stella
f=Robin

e8 e1 e7
e6 e2

e5

e9
e4 e3

By using this graph, the officer can substitute
in the handshaking theorem, incidence matrix
and adjacency matrix.

4

Handshaking
Theorem

This theorem proves that an undirected graph
has an even number of degrees and an odd
number of vertices. The undirected graph,
G=(V,E) has m vertices. This implies that
the sum of the degree of the edges in the
graph is double the number of vertices of the
graph.

5

By using the handshaking theorem, the
officer finds out that there are a total of 18
relationships. For example:

1.Moon - Lucas 10. Aria - Zoey
2.Moon - Aria 11. Aria - Moon
3.Moon - Zoey 12. Aria - Robin
4.Moon - Stella 13. Zoey - Moon
5.Moon - Robin 14. Zoey - Aria
6. Lucas-Moon 15. Robin-Moon
7.Lucas - Robin 16. Robin- Lucas
8.Lucas- Aria 17. Robin - Zoey
9. Aria-Lucas 18. Stella - Moon

These 18 relationships are knowing each
other, some are friend, lover, agent and
partner.

6

ADJACENCY
MATRIX

Adjacency Matrix by the vertices a, b, c, d, e and
f.
Adjacency Matrix is
The table shows the vertices adjacent to other
vertices of the graph.

7

[A=
a b c d ef

[da
b

c

e

f

This matrix represents the relationship in
which they know each other. 0 means that they
don't know the person while 1 means that they
know the person.

For example, column a row b is 1, which means
they know each other, got a relationship
between them. For column b row d is 0, which
means that they do not meet the person and
have no relationship among them.

8

INCIDENCE

MATRIX

Incidence Matrix is used to represent the

multiple edges. By using a column of incidence

matrix, edges are incident with the same pair of
[b
vertices.
c
e1 e2 e3 e4 e5 e6 e7 e8 e9

[da

a

e

f

Through this matrix, the officer can know that the
relationship in the case, 1 means that they have the same
relationship while 0 means that they have a different
relationship.

For instance, at row e1, column a and column b show 1.
Mean that a and b are knowing each other and have the
same memory when together. On the other hand, at row
e1, column c, d, e and f, they did not have the same
experience and the story happened the same as a and b.

9

In school time, Moon, Aria and Zoey are best
friends. One day, Moon and Aria fall in love with
Lucas at the same time. Both of them didn't tell
each other. But Lucas fell in love with Aria, they
became a couple and married after graduation.
Now, Lucas becomes a doctor while Aria becomes
a nurse. They work at the same place.

After Aria and Lucas are together, Moon becomes
a popular star.

After graduating, Zoey was unemployed. Although
Zoey has ambition like Moon, she fails when
interviewed. Zoey was very jealous of Moon but
she didn't mention it to anyone.

10

Lucas is the workmate of Moon, they
always cooperate for acting. Their acting
skills are supported by society. They get a
higher rate for their every production.

After a long time of collaboration, Lucas
falls in love with Moon. Unfortunately,
Moon just treated him as a good partner
and brother who cared very much about
her.

After getting along for a long time, Lucas
guesses that the person Moon loves is
married to her bestie, Aria.

11

Stella is the agent of Moon. She will arrange
her timetable and guide for Moon's daily
life.

Stella not only guides Moon during work
time, but she also will control Moon in her
habits such as eating habits or daily
activities. This action makes Moon feels lost
in freedom.

Moon tried to terminate the contract with
Stella. Since Moon is so popular and can
bring many benefits for the company, Stella
is unwilling to terminate the contract with
Moon.

12

SPANNING
TREE

The graph, G is a tree which is connected
and has no simple circuit graph. Spanning
tree of G means subgraph of G.

Officer decides to use the method of
Spanning Tree to observe clearly the
relationship between Moon and suspects.
From the graph, he has divided 3 subgraphs
which are shown.

By the first subgraph, the officer can
observe clearly that there is a friendship
between three girls, the love of two girls
with the same boy and the jealousy of a
friend toward her best friend.

Secondly, there is a love of a boy toward a
partner and being aware of his rival.

Lastly, there is an agent who controls the

life of a star and even intrudes on her

privacy. 13

First, he excludes two persons which are
Robin and Lucas due to they didn't have
murder motive.

Then, he suspects 3 other persons. For Aria,
her motive to kill Moon is to dislike Moon,
who loves her husband.

Next, Aria, who is jealous of Moon, wants to
kill Moon so that she can replace the
position of Moon.

Lastly, Stella although seemed not
suspicious but there's got to be reversion.
This is because when the officer collects
Moon's Room, he find that Moon wants to
kill Stella which deprives her freedom but
this action is noticed by Stella. So Stella got
the motive to kill Moon.

14

1

1 5
1 2

3 5

5 2

Officer labels the weighted graph showing
that times they contact with each other by
phone call. By converting into Kruskal's
Algorithm, officer can observe clearly their
relationship.

15

By choosing the least weight with either of
the edges, a minimum spanning tree
obtained. The method Kruskal's Algorithm is
used, which necessarily incident to vertex in
the tree and do not form circuit.

Minimum weight = 1+1+1+5+2
=10

This means that in this relationship there
are at least 10 calls.

16

SHWEHDEINE?

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suspect.
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When the time was 10pm, everyone wanted to start the
party, they noticed Moon did not appear in the living
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TIMELINE

Lucas 7pm

6pm

Sleeping

8:30pm 9pm 9:30pm

Advice Aria at room Calm down at room

9:30pm 10pm

Stay with Aria 18

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19

6pm 6:15pm Aria 7pm
Chat with Zoey
Stay at
room

8:30pm 9pm 9:30pm

Advice by Lucas at corridor Chat with Zoey

9:30pm 10pm

Stay with Lucas 20

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thheer khiutscbhaennd. . At 7pm, she had dinner with others in
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continue to quarrel with Lucas.
At about 9pm, she had a conversation with Zoey and
nthoetntsohebreefuirsraetdiotnoakl.ill MBoeocno.mShinegalasomaduvrdiseerderZoweiyll
destroy our future.
Then, Lucas found Aria and made a compromise with
htoegr.ethAetr. the time, Aria was staying with Lucas

21

6pm 6:15pm Zoey 7pm
Chat with Aria
Stay at
room

8:30pm 8:45pm 9pm 9:30pm

Stay at Toilet Chat with Aria
room

9:30pm 9:45pm 10pm

Go to Moon's Room Stay at room

22

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star.
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Then, she had a conversation with Aria. She was
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She went to Moon's room. She said at that time,
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23

Stella

6pm 6:15pm 7pm

Chat with Stay at room
Moon

8:30pm 9pm 9:30pm

Calling Stay at room

9:30pm 9:45pm 10pm

Stay at room Go to Moon's Room

24

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At about 9.45pm, she went to find Moon and
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carormivbeidneadt Mthoeonp'osisRoonomw,itMhotohneddieridn.kS. hBeutdiwdhneont
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25

Robin 7pm

6pm 6:15pm

Stay at room Chat with Moon

8:30pm 8:45pm 9pm 9:30pm

Toilet Stay at Strolling
room

9:30pm 9:45pm 10pm

Strolling Stay at room

26

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After that, he went to the toilet and met Zoey. At
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plan.

27

At the time slot 9:30pm to 9:45pm, Zoey did not
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stayed in her room.

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cwoitlohuLr.ucas and Robin, she is represented in pink

This means that in this time slot, Zoey is more
doubted.
28

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29

HHOOWWIISS
TTHHEE

MMUURRDDEERR??

By restore the murder process, the
officer draws an architectural plan floor
plan. He drew a Deluna Homestay and
specified the room. Also, he draws the
doors and windows. The plan is shown.

Next, he draws a graph and wants to find
the shortest path for how Zoey kills
Moon.

30

31

14m

Zoey 6m Kitchen
3m 7m 4m
Robin
Lucas
and Aria

4m 3m
Moon
5m
Stella

From the graph, the officer needs to calculate the
distance from Zoey's room to the kitchen and the last
destination to Moon's room with the shortest
distance.
1. Zoey - Kitchen-Moon =6+7

= 13m
2. Zoey - Kitchen - Moon= 6+14

= 20m
3. Zoey- Kitchen - Robin - Stella-Moon=6+4+3+5

=18m
4. Zoey- Kitchen - Zoey - Lucas&Aria = 6+6+3+4

= 19m

32

There are 4 paths for Zoey to go to the kitchen and
Moon's room.

First, in Deluna Homestay, the distance from Zoey's
room to the kitchen is 6m and then to Moon's room
is 7m. So the distance of the first path is 13m.

Secondly, Zoey's room to kitchen also the same,
which is 6m but she needs to go outside and climb
through the window into Moon's room which is
distance is 14m. And the total distance is 20 m which
is the longest distance.

Next, the distance from Zoey's room to the kitchen
and then to Robin's room and Stella's room is 6m,
4m and 3m respectively. Zoey needs to pass through
their room then arrive at Moon's room. The total
distance is 18m.

Lastly, Zoey goes to Kitchen and back to her room,
and then passes through Lucas and Aria's room. This
will take about 19m.

So Zoey chose the shortest path which is from her
room to the kitchen and Moon's room.

33

The officer concludes that Zoey went to the
kitchen to get the knife and walk toward
Moon's Room. She went to find Moon at
9:30pm and then used the reason to have a
chat for entering her room. After entering,
Zoey knocked down Moon who did not alert
Zoey. Then, Moon was unconscious and
was dragged to bed.

After that, Zoey aimed at Moon's heart and
stabbed the knife into it. Moon bleeds and
dies.

Zoey had done many precautions such as
bringing the gloves to prevent the
fingerprints from being scanned. Also, she
used the alternative way by taking the
shortest path toward the Moon's room.

The officer arrests Zoey and closes her into
the prison hoping that she can regret it.

34

Through this zine, I had a lot of brain
storming. First, I think of the scenario
and then I need to create a story. After
that, I need to link the character, time
and relationship in my storyline. This is
quite challenging for me.

Besides, I also link all the scenarios
with the mathematical form. I learned
that mathematics is very useful in our
daily life.

The scenario that I created lets the
reader think about who is the
murderer. And can solve the case
together with the "officer". This is quite
interesting and fun.

Kenneth H. Rosen (2019). Discrete Math And Its
Application.

Adarsh Nagarajan, Application of Minimum Spanning

Tree algorithm for network reduction of distribution

systems,

https://www.researchgate.net/publication/28667944

9_Application_of_Minimum_Spanning_Tree_algorithm

_for_network_reduction_of_distribution_systems ,

September 2014

Applied AI Course, Facebook Friend Recommendation

Using Graph Mining.

https://www.youtube.com/watch?v=vMR0E6tJVrA,2

June 2021.

Mohammad Ashikur Rahman, Application of Graph
Theory in Google Maps,
https://www.youtube.com/watch?v=_AFqdsqIs-A , 1
June 2021.

Rhyd Lewis, Applications of Graph Colouring,
https://www.youtube.com/watch?v=y4RAYQjKb5Y , 3
June 2021.

Science Please, How To Solve A Crime With Graph

Theory, https://www.youtube.com/watch?

v=TwHy2DuWB3k , 3 June 2021.