NETWORK IN GRAPH THEORY (EDITED TXT BOOK)

CHAPTER

Network in

5 Graph Theory

You will learn
etwor

The transportation system in alaysia which comprises land,
water and air transportation are developed in line with the
country s progress The land transportation system, particularly
public transport is developed at a fast pace based on the increase
in the number of commuters in big cities City rail services such as
Light Rail Transit LRT , ass Rapid Transit RT , monorails
and commuter trains in the cities are amongst the types of public
transportation that are increasingly well accepted by the public

o you now that the transportation system is a type of networ

Why Study This Chapter?

networ is used to lin ob ects in the same field based on its
needs etwor s are widely used in transportation, computer,
social, business, investigation, medicine, science, neuroscience,
social science and gaming fields
128

Walking Through Time

WORD BANK Leonhard Euler

• weighted er e erat
• vertex Leonhard Euler, a mathematician in the th
• degree dar a century, was born in asel, wit erland n the
• discrete d s ret year , Euler solved a mathematics and
• loop el logic problem nown as the even ridges of
• graph ra
• simple nigsberg and developed a mathematical structure
• tree da called graph a diagram consisting of dots
• networ vertices which are lin ed by lines or arcs edges
• subgraph ra a a
• edge s ra http bt sasbadi com m
• directed te
terara

129

Chapter 5 Network in Graph Theory

5.1 Network

What is the relationship between a network and a graph? Learning
Standard
n ear , you were introduced to systems of computer networ s
and the world of nternet through the sub ect of nformation and To identify and explain
Communication Technology a network as a graph.

The lin between a group of computers and the associated devices,
that is a computer networ , enables information to be searched, used
and shared easily o you now the relationship between a networ and
a graph

graph is used to represent data consisting of discrete ertex

ob ects and to show the relationship between these ob ects in

a simple graphical manner n the field of mathematics, graph Edge

theory in particular, a graph is interpreted as a series of dots

which are either lin ed or not lin ed to one another by lines

Each dot is nown as a vertex and the line oining two vertices

is nown as an edge

CHAPTER 5 graph is usually used to represent a certain networ network is part of a graph with
the vertices and edges having their own characteristics The structure of networ data has
a many to many relation Examples of graphs that involve networ s are as follows

Land transport network

Vertex
Regions, towns, cities or certain buildings that
are lin ed

Edge
Roads, highways or railway lines

Social network INFO ZONE

Vertex A Cold Chain System is a
ndividuals, groups or system used in the world
organisations of medicine. The function
of this system is to
Edge transport, distribute and
Types of relationships store vaccine and blood
such as friends, in a fixed temperature
colleagues or families range from the source to
the place they are used.

130

Chapter 5 Network in Graph Theory

A graph is denoted by a set of ordered pairs G = (V, E), where INFO ZONE
• V is the set of dots or vertices.
G = Graph
V = {v1, v2, v3, ... vn} v = Vertex or dot
e = Edge or line or arc
• E is the set of edges or lines linking each pair of vertices. d = Degree
E = {e1, e2, e3, ... en} ∑ = Sum
E = {(a1, b1), (a2, b2), ... (an, bn)}; a and b are pairs of vertices.

The degree, d, of a vertex, v, is the number of edges that is connected MY MEMORY
to other vertices. The sum of degrees of a graph is twice the number ∈ : an element of
of edges, that is

Σd(v) = 2E ; v ∈ V

What do you understand about a simple graph?

A simple graph has no loops and no multiple edges. The sum of degrees of the graph is twice the
number of edges.

1 12 CHAPTER 5

Based on the simple graph given, determine 6

(a) V and n(V) 3
(b) E and n(E)
(c) sum of degrees.

54

Solution: INFO ZONE

(a) V = {1, 2, 3, 4, 5, 6} Set of vertices The edge for the vertex
n(V) = 6 Number of vertices pair (1, 2) is also the
edge for the vertex pair
(b) E = {(1, 2), (1, 5), (2, 3), (2, 4), (3, 4), (4, 5), (5, 6)} Set of vertex pairs (2, 1).

n(E) = 7 Number of edges

(c) Sum of degrees Degree of vertex: d(1) = 2 The degree of vertex 1 is
Σd(v) = 2(E) d(2) = 3 two, that is the edges which
= 2(7) d(3) = 2 connect vertex 1 to vertex 2
= 14 d(4) = 3 and vertex 1 to vertex 5.
d(5) = 3
Example 2 d(6) = 1
Sum = 14

State the number of vertices, edges and the sum of degrees for the following simple graphs.

(a) 1 2 3 4 (b) B C D

A

765 GF E

131

Chapter 5 Network in Graph Theory

Solution: b umber of vertices
umber of edges
a umber of vertices um of degrees × umber of edges
umber of edges ×
um of degrees × umber of edges
×


What is the meaning of multiple edges and loops of a graph?

ultiple edges Edge
P
• nvolve two vertices ertex Q R
• The vertices are connected by more than T S Loop

one edge

• The sum of degrees is twice the number

of edges

Loops ultiple edges between
vertex P and vertex

CHAPTER 5 • nvolve one vertex
• The edge is in the form of an arc that starts and ends at the same vertex
• Each loop adds to the degree

Let the graph given be denoted by a set of ordered pairs, G V, ,
then

V A, , , } e

ultiple edges loop

E A, , A, , B, , B, , , , B, , A, , , e ee

E e , e , e , e , e , e , e , e } ee

irst edge of AB econd edge of AB A Be
e

Example 3 P MY MEMORY
V = Set of vertices
The diagram on the right shows a graph with a loop and , , , , E = Set of edges
multiple edges tate , ,
a V and V
b E and E
c sum of degrees

Solution:
a V P, , , , , }
V

Saiz sebb e nE ar P, , P, , P, , , , , , , , ,

E

132

Chapter 5 Network in Graph Theory

c um of degrees TIPS

egree of vertex P The sum of degrees is The degree of a vertex
egree of vertex with a loop in an undirected
egree of vertex graph is , one in cloc wise
egree of vertex direction and the other in
egree of vertex anticloc wise direction
egree of vertex

Example 4

raw a simple graph based on the given information

a V , , , , b V P, , , , , }

E , , , , , , , , , , , E P, , P, , , , , , , , ,

Solution: b P QR
a

Example 5 CHAPTER 5

raw a graph with multiple edges and loops based on the given information

a V P, , , }
E P, , P, , P, , , , , , , , , , ,

b V , , , , , ,
E , , , , , , , , , , , , , , , , , , , , , , ,

Solution: b TIPS
a
• The vertex pair for a loop
P is in the form a, a

• The vertex pairs for
multiple edges are in the
forms a, , a,

Example 6

etermine whether a graph with the following degrees of vertices can be drawn

a , , , , b , , , , ,

Solution: b um of degrees

a um of degrees
The graph can be drawn because Saiz sebenar
the sum of degrees is even
The graph cannot be drawn because
the sum of degrees is odd

133

Chapter 5 Network in Graph Theory

5.1a

1. Three simple graphs are given below S c A BC
a 1 e 2 e 3 b Q R T

e e ee

5e 4 P FE D
W
VU

or each graph, determine

i V and V ii E and E iii sum of degrees

2. Two graphs with multiple edges and loops are given below

a A B b P


CHAPTER 5 O
E

or each graph, determine

i V and V ii E and E iii sum of degrees

3. raw a simple graph based on the given information
a V , , , , ,
E , , , , , , , , , , , , ,

b V P, , , , , }
E P, , P, , , , , , , , , , , , ,

4. raw a graph with multiple edges and loops based on the given information
a V P, , , }
E P, , P, , , , , , P, , , , , , , , P, , ,

b V , , , ,
E , , , , , , , , , , , , , , , , , , , , , , ,

5. raw a simple graph with the given degrees of vertices

a , , , , b , , , , ,

6. raw a graph with loops and multiple edges with the given degrees of vertices

a , , b , , , ,

134

Chapter 5 Network in Graph Theory

What is the difference between a directed graph and Learning
an undirected graph? Standard

A directed graph is a graph in which a direction is assigned to the edge Compare and contrast
connecting two vertices. Directed graphs are usually used to represent directed graphs and
the flow of a certain process. For example, road maps, airlines networks, undirected graphs.
electrical circuits, computer networks and organisation charts. weighted graphs and
unweighted graphs

B INTERACTIVE ZONE

> The blood circulation
system in the human
A body is also categorised
C as a directed graph.
Why?
Diagram 1
> >>
>>

CHAPTER 5
Diagram 1 shows a simple directed graph. Based on the directions of INTERACTIVE ZONE
the arrows, it can be seen that,
For electricity networks,
(a) for edge AB, A is the initial vertex and B is the terminal vertex. power stations,
transformer stations,
(b) for edge CB, C is the initial vertex and B is the terminal vertex. substations and
consumers are the
(c) all vertices are connected in one direction only. vertices, and the cables
and wires are the edges.
P> e1> Q e2 Is an electricity network
e5 e4 a directed graph or
R e3 an undirected graph?
Discuss.

Diagram 2

Diagram 2 shows a directed graph with a loop and multiple edges. Based on the directions of the
arrows, it can be seen that

(a) e4 = (P, R); P is the initial vertex and R is the terminal vertex.
(b) e5 = (R, P); R is the initial vertex and P is the terminal vertex.
(c) e2 = (Q, Q); Q is the initial vertex and the terminal vertex because e2 is a loop.

An undirected graph is a simple graph or a graph with loops and multiple edges drawn without
any direction being assigned.

135

Chapter 5 Network in Graph Theory

Differences between directed graphs and undirected graphs

Type of Graph Set V and Set E Degrees
graph
Undirected graph V = {A, B, C, D} d(A) = 2, d(B) = 2,
E = {(A, B), (A, C), d(C) = 3, d(D) = 1
AD Σd(V) = 8
(B, C), (C, D)}

BC The order of the vertices written
is not important. Both pairs
Directed graph of vertices, (A, B) and (B, A)
represent the edge AB.
AD
Simple V = {A, B, C, D} din (A) = 2 and
graph B >C E = {(B, A), (C, A), dout (A) = 0
CHAPTER 5 > Thus, d(A) = 2 + 0
Undirected graph > (B, C), (D, C)}
> d(A) = 2
>R> The order of the vertices
PQ are written according to the din (A) means the number
direction of the edge. (B, A) and of edges ‘going into’
(A, B) represent different edges. vertex A.
dout (A) means the
number of edges ‘coming
out’ from vertex A.

V = {P, Q, R, S, T} din (B) = 0, dout (B) = 2
din (C) = 2, dout (C) = 1
E = {(P, T), (P, Q), (Q, R), din (D) = 0, dout (D) = 1
(Q, R), (Q, S), (S, T), Σd(V) = 8
(T, T)}
d(P) = 2, d(Q) = 4
d(R) = 2, d(S) = 2
d(T) = 4
Σd(V) = 14

Graph with TS V = {P, Q, R, S, T} din (P) = 1, dout (P) = 1
loops and din (Q) = 3, dout (Q) = 1
multiple Directed graph E = {(P, Q), (Q, R), (R, Q), din (R) = 1, dout (R) = 1
(S, Q), (T, S), (T, T), din (S) = 1, dout (S) = 1
edges R (T, P)} din (T) = 1, dout (T) = 3

136 P > Q>

T> S
>
>

}

}
Loop = 1 Loop = 1
(T, P) = 1
(T, S) = 1

Σd(V) = 14

Chapter 5 Network in Graph Theory

What are the differences between weighted graphs and unweighted graphs?

Type of Weighted graph Unweighted graph
graph
Edge Directed graph and undirected graph Directed graph and undirected graph

Example Associated with a value or a weight Not associated with a value or a weight

The edge represents: The edge relates information like:
• distance between two cities. • job hierarchy in an organisation chart.
• travelling time. • flow map.
• the current in an electrical circuit. • tree map.
• cost. • bubble map.

Example 7

Draw a directed graph based on the given information. INFO ZONE

(a) V = {P, Q, R, S, T, U} (b) There is a loop at vertex Q and For Example 7(a), the CHAPTER 5
E = {(P, Q), (P, R), RS is a multiple edge such that vertex U appears in
(R, Q), (S, R), set V but not in set E.
(S, Q), (S, T)} din (P) = 1, dout (P) = 1 This means vertex U is
din (Q) = 3, dout (Q) = 2 not connected to any
din (R) = 0, dout (R) = 3 other vertices in the
din (S) = 3, dout (S) = 1 graph and it is known
as an isolated vertex.

Solution: Direction
}P to Q
(a) Vertex pair Two edges from vertex P P>R
(P, Q) P to R
(P, R)
(R, Q) R to Q One edge from vertex R > >
(S, R) Three edges from vertex S >>
(S, Q) }S to R Q > S
(S, T) U > T
S to Q
S to T

Vertex T only connected to vertex S

(b) Total number of vertices = 4 RS – a multiple edge, Complete the graph based on the
A loop at vertex Q. number of edges going into and
PQ coming out from each vertex.
PQ
>
P >Q

> >
>

SR SR S> R

137

Chapter 5 Network in Graph Theory

Example 8 tart
v

The diagram on the right shows one way paths that arul can m
v
choose for his running practice ertex v is the starting position
CHAPTER 5
m
and vertex v is the ending position before he goes home m
m
etermine

a the shortest distance from v to v v m
b the longest distance from v to v
v v
c the vertices that must be passed through if the distance of the m End m

one way run is between m and m

Solution: b Longest distance c v , v , v , v
a hortest distance v v v v v and
v v v m v , v , v , v
m m
m m
m

5.1b

1. tate two differences between directed graphs and undirected graphs

2. hat is the meaning of weight in a weighted graph

3. raw a directed graph based on the given information

a V P, , , , , , } iii d P ,d t P
E P, , , , , , , , , , , , , d , d t
,d t
b i There is a loop at vertex and a loop at vertex d ,d t
ii is a multiple edge d

4. a ased on the information in Table and Table , complete the weighted and undirected graph

Name of place Vertex Vertex pair Weight (km)
uala Pilah KP J, B
ahau B JB
Rompin KP, KP
atu i ir BK B,
uasseh J B, BK ncomplete graph
BK,
KP,

Table Table

b r enny and r uruges drive individually from uala Pilah to Rompin such that

i r enny uses the shortest route
ii r uruges ta es the route which passes through uasseh and ahau
Calculate the difference in distance, in m, for the ourneys ta en by r enny and r

uruges from uala Pilah to Rompin

138

Chapter 5 Network in Graph Theory

How do you identify and draw subgraphs and trees? Learning
What do you understand about a subgraph? Standard

A subgraph is part of a graph or the whole graph redrawn. A graph H Identify and draw
is said to be a subgraph of G if, subgraphs and trees.

(a) the vertex set of graph H is a subset of the vertex set of graph G, that is V(H) ʚ V(G).
(b) the edge set of graph H is a subset of the edge set of graph G, that is E(H) ʚ E(G).
(c) the vertex pairs of the edges of graph H are the same as the edges of graph G.

In short, INFO ZONE
• a vertex in graph G is a subgraph of graph G.
• an edge in graph G along with the vertices it connects The symbol ⊆ also

is a subgraph of graph G. stands for subset.
• each graph is a subgraph of itself.

Example 9

Determine whether Diagram 1, Diagram 2, Diagram 3 and Diagram 4 are the subgraphs of graph G.

P e1 Q e2 P e2 P e1 Q P Q e2 P e1 Q e3

e5 e3 e5 e5 e3 e5 e2 CHAPTER 5

S e4 R S S S S e4 R
Graph G Diagram 1 Diagram 2 Diagram 3 Diagram 4

Solution:

Diagram 1 – Yes, because the vertex pair for edge e5 is the same.
{e5} {e1, e2, e3, e4, e5} and {P, S} {P, Q, R, S}

Diagram 2 – No, because the position of loop e2 is not on vertex Q.
Diagram 3 – No, because the edge connecting vertex P and vertex S is not e3.
Diagram 4 – No, because the loop and the edge connecting vertex Q and vertex R are wrong.

What do you understand about a tree? INFO ZONE

A tree of a graph is a subgraph of the graph with the following The term tree was
properties: introduced by Arthur
Cayley, an English
(a) A simple graph without loops and multiple edges. mathematician, in the
year 1857.
(b) All the vertices are connected and each pair of vertices is
connected by only one edge. INFO ZONE

(c) Number of edges = Number of vertices – 1 A family history chart is
Number of vertices = n an example of a tree.
Number of edges = n – 1
139

Chapter 5 Network in Graph Theory iagram is a tree because

A • all the vertices are connected
E • every pair of vertices is connected by an edge only
• there are no loops or multiple edges
B • vertices, edges

CD
iagram

A iagram is not a tree because
E
• vertex B and vertex can be connected in two ways
B i
ii
CD
iagram • vertices, edges

CHAPTER 5 Example 10 Q R

etermine whether each of the following graphs is a tree ustify your answer
a P Q R b P Q R c P Q R d P



UT S UT S UT S UT S

Solution:

a ot a tree vertices, edges Each pair of vertices can be connected in various ways
b tree vertices, edges Each pair of vertices is connected by one edge
c tree vertices, edges Each pair of vertices is connected by one edge
d ot a tree vertices, edges ertex and vertex are not connected to the other vertices

Example 11

raw a tree for the following information given

a vertices b vertices c edges d edges

Solution: d edges
vertices
a vertices b vertices c edges
edges edges vertices

Saiz sebenar

140

Chapter 5 Network in Graph Theory

Example 12

Draw two trees based on the graphs given below. INFO ZONE
(a) (b)
Trees are used to
Solution: determine the shortest
path with the condition
that the path passes
through each vertex
once only.

(a) 5 vertices (b) 6 vertices
7 edges (exceed by 3) 8 edges (exceed by 3)

(i) (ii) (i) (ii)

Example 13

The following diagram shows an undirected weighted graph. Draw a tree with a minimum CHAPTER 5

total weight. Q 20 R

25 10 12 17

P 19 T 14 S

Solution: LanSgtkeaph22
• Between the weights 19 and 25, keep
Step 1
weight 19 because its weight is smaller.
5 vertices, 7 edges • Between the weights 12, 14 and 17,
• 3 edges to be removed.
remove weight 17.
• Remove edges with the greatest weights
(PQ, QR, PT) QR

QR

10 12 17 10 12

P T 14 S P 19 T 14 S

The graph above is not a tree because The graph obtained is a tree.
• vertex P is not connected to the other vertices. Minimum total weight of the tree
• three edges, RS, ST and RT, connect three = 10 + 12 + 14 + 19
= 55
vertices only.

141

Chapter 5 Network in Graph Theory

5.1c

1. etermine whether the given diagrams are the subgraphs of graph G

e e P e P
P P P
e e e

e iagram iagram iagram iagram iagram
raph G P e e e e

e e e P e e e P
P e P e e
e P e e
e iagram e iagram e
iagram iagram iagram
iagram

2. raw five subgraphs for each given graph

a P b P e S c P e e
R
CHAPTER 5 e e e e e
e e

Qe R Q e R Q d

3. dentify whether it is a tree or not a tree for the following diagrams

a b c

4. raw a tree based on the given information

a vertices b vertices c edges d edges

5. raw two trees based on the given graphs

a b

6. The diagram on the right shows an undirected Q R
weighted graph 18
24 38 S
a raw a tree with a minimum total weight P 20 U 36
b hat is the minimum total weight
32
142
34 42
T 30

Chapter 5 Network in Graph Theory

How would you represent information in the form of networks?

networ is a type of unique graph and can be used to represent Learning
overlapping and intersecting information etwor s are widely used Standard
in almost every area of our daily lives networ that is drawn and
displayed in graphic forms enables the interrelationships between Represent information
various information or data structures to be understood easily in the form of networks.

etwor s can be drawn as CHAPTER 5
• directed weighted graph or directed unweighted graph
• undirected weighted graph or undirected unweighted graph


Transportation Networks

Transportation networ s can be shown as
weighted graphs and unweighted graphs
The weights can represent the distance,
travelling time or cost of the ourney The
well nown navigation system in the weighted
transportation networ s is the P lobal
Positioning ystem
The diagram on the right shows the train
transit networ in uala Lumpur entral

L entral This undirected graph is an
example of transportation networ s with the
vertices representing names of stations that are
connected and the edges representing the types
of trains

Example 14

r oon and his family plan to visit historical

places in ela a The map shows three

alternative routes with the distances and

estimated times needed to travel from Tang a

to amosa, ela a

ssume P is a ilometre route, is a

ilometre route and is a ilometre

route n your opinion, why does route P ta e

the longest time compared to the other routes

even though route P is the shortest route

Solution:

Route P ta es the longest time because the route passes through crowded town areas and there are
more road users compared to the other two routes

143

Chapter 5 Network in Graph Theory

Social Networks

ocial networ s are becoming more popular among teenagers and Smart Mind
adults ocial networ s are used in areas li e ob opportunities,
business opportunities, socialising, family relationships, education, State a social network
social media and connecting with communities around the world that you know of.
Even though social networ s are main platforms for various
activities and are useful, you should be cautious and moderate in using INTERACTIVE ZONE
social networ s to avoid being distracted and being deceived easily

Discuss the negative
effects of using social
networks.

Example 15

The table below shows the data of six pupils and the games that they li e Represent the information
in the form of a networ

CHAPTER 5 Name of pupil Game Smart Mind
Edmund adminton, Chess
wan ootball, e a ta ra What are the meanings
Ra an Chess, ootball of LTE and 4G which
ina Chess, etball are often used in
wireless Internet
networks?

aria adminton, etball INTERACTIVE ZONE
enny etball, olleyball
Solving criminal cases is
Solution: also related to graphs.
All evidence obtained are
Let the vertices represent types of games and the edges linked to one another.
represent the pupils names Discuss information that
can be represented by
the vertices and edges
of graph.

Chess Ra an ootball wan e a ta ra
adminton ina
Edmund etball TIPS
aria
enny Choose information that is
not repeated as the edges
olleyball
INTERACTIVE ZONE
Is the food chain a
network? Discuss.

144

Chapter 5 Network in Graph Theory

5.1d

Name of place Vertex Vertex pair Distance (km) MK
,
uala epetang iagram
K,
atang M K,

amunting K ,
,
impang ,
,
Taiping ,
,
Chang at

Table

Table

1. Table and Table show names and distances of six places in Pera iagram shows an CHAPTER 5
incomplete undirected graph connecting the six places

a Complete iagram by drawing an undirected weighted graph
b raw a tree with a minimum total weight which shows every place being visited once only
c hat is the minimum distance of the tree that you have drawn

2. The table below shows four types of favourite food of several pupils

Food Name of pupil
Chic en rice ervin, Ra , elen, ong, in
ervin, urul, tiqah, in, Puspa
as le a elen, ulia, urul, aru , Puspa
ried rice aru , tiqah, Ra , ong, ulia
ried noodles

a ased on the table, draw a graph with multiple edges

b etween the types of food and the names of pupils, which group will you use to represent
the vertices ustify your answer

c hat is the relationship between the sum of degrees of your graph with the total number
of food choices

d etween table form and graph form, which form is clearer in showing the relationship
between the types of food and the pupils

3. a raw an organisation chart of your class using your own creativity

b tate the type of graph you have used s the class organisation chart a networ ustify
your answer

145

Chapter 5 Network in Graph Theory

How do you solve problems involving networks? Learning
Standard
Example 16
Solve problems
The table below shows the choices of public transportation, estimated involving networks.
travelling time and estimated cost for a ourney from ohor ahru
to ota haru

Type of Travelling time Estimated Estimated expenses
transportation (24-hour system) duration of
Price of ticket per person/cost of fuel
us hours journey
R R
Train hours
ith bed
Cab hours hours Child R R

CHAPTER 5 hours dult R R

ithout bed
Child R

dult R

R for one cab

ased on the table, determine the type of transportation that should be chosen for the situation
given below ustify your answer

a ourney involving an adult without time constraint

b ourney involving an adult with time constraint

c ourney involving two adults and two children

Solution:

Type of transportation Price of ticket per person (RM)

Minimum Maximum

us Child Child
Train ith bed dult dult

Train ithout bed Child
dult

a Ta ing a train with bed is the best choice because the difference in price is only R compared
to ta ing a train without bed The passengers can have a good rest throughout the whole ourney
This choice is also cheaper compared to ta ing a cab

b Ta ing a bus is the best choice because the duration of ourney is shorter than that of a train and
it is more economical than ta ing a cab or safety purpose, it is not wise for an individual to
ta e a cab for a long ourney

c Ta ing a cab is the most economical choice

146

Chapter 5 Network in Graph Theory

Example 17

The map above shows the domestic flight routes of a private airline in alaysia CHAPTER 5
a r oshua wor s in uala Lumpur e wants to visit his family in ota inabalu tate the

best route r oshua can choose
b hat are the advantages and disadvantages for the choice of flight made

Solution:

a The direct flight from uala Lumpur to ota inabalu

b The direct flight from uala Lumpur to ota inabalu saves time and cost The flight from
uala Lumpur to ota inabalu with transit ta es a longer time and most probably the cost of

the ourney is also higher

5.1e

1. r aswi wor s in ulai e plans to visit his family in iri on a certain wee end On riday,
r aswi s wor ends at noon The table below shows the choices of domestic flight

routes of a private airline in alaysia on riday and aturday for that wee

Route Friday Saturday

ohor ahru Time Price of ticket Time Price of ticket
iri
o flight hrs R
ohor ahru
uching hrs R hrs R
hrs R hrs R
uching hrs R hrs R
iri hrs R hrs R

a etermine the most economical flight from ohor ahru to iri

b etermine the best flight that r aswi can choose if he needs to go bac to Peninsular
alaysia on unday ive your reason

147

Chapter 5 Network in Graph Theory

1. Three graphs are given below

a P Q b c Q R
T
R U

P
R
U

TP S

TS QS

or each graph, determine

i set . ii set . iii sum of degrees

CHAPTER 5 2. raw an undirected graph based on the given information
a V P, , , , , }
E P, , , , , , ,

b V P, , , , }
E P, , P, , , , , , , , , , ,

3. a raw an undirected weighted graph to Name of town Distance (km)
represent the federal roads connecting a Temerloh enta ab
few towns as shown in the table ou are Temerloh andar era
advised to use an area map to determine
the positions of the towns andar era Teriang
enta ab Lanchang
b s the graph that you have drawn a tree Lanchang ara
ustify your answer enta ab uala rau

4. The directed graph on the right shows the m m m
roads connecting Lani s house at A to the E m A
school at E The edge BE is a municipal road chool
and edges AB, , and are roads in m
the housing estates uggest the best route B
that Lani can choose to cycle to school
ustify your answer

148

Chapter 5 Network in Graph Theory

5. The directed weighted graph on the right shows PR hours
the prices of tic ets and the travel times for some hours
choices of flights of a private airline ertex is the R
destination of the flight from vertex P ertex and R hour
vertex are the transit airports The transit time at
each of the airports is minutes hour
R
a tate hours

i the most economical route R

ii the route that ta es the shortest time R hours

b f you need to go to a destination at optimum cost, state the route that you will choose
ustify your answer

6. rite five linear equations based on the directed graph below TIPS
iven x , determine the values of x , x , x and x
se Σdin Σdout for each
AB vertex

x x m
x x m

xE CHAPTER 5

7. The following undirected graph shows six houses in a village salesperson needs to visit all
the houses starting from house A and finishing at house F

F

m

m m E
A m m

m m
m

B

a raw a directed graph to represent the shortest distance from A to with the condition that
all the paths are ta en once only

b ased on your graph, calculate the shortest distance in m

149

Chapter 5 Network in Graph Theory P G A
X L YM
8. The graph on the right shows the connections J
between the elements in set X, set Y and set Z, B F
where the universal set, X ʜ Y ʜ Z Z E
H
a Represent the graph in a enn diagram N
b etermine
i X ʜ Y
ii X Z Y
iii Y Z X Y

IK

9. r anesan is the manager of an insurance agency e M
A
recruites two active insurance agents to sell the latest insurance

scheme valued at R per month esides selling insurance

policies, each agent needs to recommend at least two new A

insurance agents n the incomplete tree on the right, M

represents the manager, A represents the first level agents and

CHAPTER 5 A represents the second level agents A

a f there are agents in anuary, complete the given tree

b The table on the right shows the percentages of basic gent
commission received by an agent and the manager for anager

an insurance policy sold

i C alculate the total basic commission received by
r anesan in anuary if the minimum number of

insurance policies sold by an agent in anuary is

ii hat is the minimum number of policies that an agent needs to sell in order to
receive a basic commission of at least R

10. The diagram on the right shows the growth of a type of cell
t is given that on the first day, there are four cells On the
second day, each cell produces three cells On the following
day, each new cell produces another three new cells The
process of producing new cells repeats at the same rate

a O n which day will the total number of cells exceeds
for the first time

b Calculate the total number of cells on the fifth day

c iven the life span of a cell is days, calculate the total
number of cells on the eighth day

150

Chapter 5 Network in Graph Theory

PROJ ECT

1. ma e is an example of a networ
iagram shows an example of a ma e whereas iagram is the corresponding

networ

iagram CHAPTER 5

IP QR

O NX V
J

HK

L MWS

E D TU

FG YZ

ABC

iagram

2. ivide the class into groups
3. Obtain examples of ma es with different levels of difficulty
4. Label the vertices with suitable letters as shown in iagram
5 raw the corresponding networ s
6. Exhibit your pro ect wor at the athematics Corner

Saiz sebenar

151

Chapter 5 Network in Graph Theory

CONCEPT MAP

Network in Graph Theory

Simple Graph Has loops and multiple edges
Degree

Undirected graph Directed graph

CHAPTER 5
favourite

time
Subgraph Weighted graph cost

QQ distance
name
PR PR
Q Q Unweighted graph
R
P Tree (Simple graph)
Q P
Q
R
Q R

PR PR
Q Q

PRPR

152

Self Practice 4.3c (b) (i) V = {P, Q, R, S, T, U, V, W}

1. 39 n(V) = 8
2. x = 4
3. 12 (ii) E = {(Q, P), (Q, R), (Q, W), (R, V), (S, T), (S, U),
4. (a) 41
(U, V), (V, W)}
(b) 25
(c) 7 n(E) = 8

(iii) 16

(c) (i) V = {A, B, C, D, E, F}

n(V) = 6

(ii) E = {(A, B), (A, F), (B, C), (B, E), (C, D), (C, E),

(D, E), (E, F)}

n(E) = 8

1. (a) P ʝ Q = {3, 5} (iii) 16

(b) P ʝ R = {3} 2. (a) (i) V = {A, B, C, D, E}

(c) P ʝ Q ʝ R = {3} n(V) = 5

(d) (P ʝ Q ʝ R)' = {2, 5, 6} (ii) E = {(A, B), (A, B), (A, E), (B, C), (B, D),

2. (a) M ʜ N = {a, b, d, i, k, u} (B, E), (C, C), (C, D), (D, E), (D, E)}

(b) M ʜ P = {a, b, e, i, k, n, r} n(E) = 10

(c) M ʜ N ʜ P = {a, b, d, e, i, k, n, r, u} (iii) 20

3. (a) PQR (b) P R (b) (i) V = {O, P, Q, R, S, T, U}
Q
n(V) = 7

(ii) E={(P,U),(P,U),(U,T),(U,T),(P,Q),(P,O),

(Q, R), (Q, R), (Q, O), (R, R), (R, S), (R, S),

PʝQ PʜR (R, O), (S, O), (S, T), (T, O), (U, O)}

4. (a) T' = {1, 3, 5, 6, 8} n(E) = 17

(b) S ʜ T = {2, 4, 5, 6, 7, 8, 9} (iii) 34

(c) S' ʝ T = {2, 4, 9} 3. (a) (b)

(d) (S ʜ T)' = {1, 2, 3, 4, 5, 6, 8, 9} 12 3 4 P Q R

5. A' = {d, e, f, h, i}

6. (a) Q' = {11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24,

26, 27, 28, 29} 6 5
(b) U T S
(b) P ʜ R' = {10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 4. (a) 2

21, 22, 23, 25, 26, 27, 28, 29, 30} Q
1 53
(c) (P ʜ R)' ʝ Q = {10, 15, 20, 25, 30} P
R
7. (a) B C (b) A B C
4
A (b)

S

A ʝ (B ʜ C) C ʜ (A ʝ B)' 5. (a)

8. 39 (b) 11 (c) 54 (d) 2 6. (a) (b)
9. 31 (b) 5 (c) 7
10. 6
11. (a) 8
12. (a) 8
13. 50

CHAPTER 5 Network in Graph Theory

Self Practice 5.1a Self Practice 5.1b

1. (a) (i) V = {1, 2, 3, 4, 5} 1. (i) The edges in directed graphs are marked with
n(V) = 5 direction.

(ii) E = {(1, 2), (1, 5), (2, 3), (2, 4), (2, 5), (3, 4), (ii) The order of vertices in directed graphs are
(4, 5)} written according to the direction of the edges.

E = {e1, e2, e3, e4, e5, e6, e7} 2. A value or information involving edge.
n(E) = 7
(iii) 14 299

3. (a) P ST (b) P (c) Sum of degrees
QS = Number of pupils × Total number of food choices

Q R UV R (d) Graph form
4. (a) 3. (b) Undirected graph. The organisation chart is
BK (b) 1.6 km
KP 6.6 a network because it shows the relationships
11.4 between the individuals involved based on the
chart’s requirement.
9.3 11.6 B
J Self Practice 5.1e

20.7 1. (a) Johor Bahru – Kuching (Saturday, 0605 hours)
40 and then Kuching – Miri (Saturday, 1145 hours).

R (b) Johor Bahru – Kuching (Friday, 1930 hours) and
then Kuching – Miri (Friday, 2155 hours). Even
Self Practice 5.1c though the total price of the flight tickets is
RM35 higher than the cheapest package on
1. Subgraph – Diagram 1, Diagram 2, Diagram 3, Saturday, Encik Maswi gets to spend more time
with his family.

Diagram 4, Diagram 8, Diagram 10,

Diagram 11

Not a subgraph – Diagram 5, Diagram 6, Diagram 7, 1. (a) (i) V = {P, Q, R, S, T, U}

Diagram 9 (ii) E = {(P, Q), (P, S), (P, U), (Q, R), (Q, T), (R, S),

3. (a) Not a tree (b) Not a tree (R, U), (S, T), (T, U)}

(c) Tree (d) Not a tree (iii) 18

6. (a) Q (b) (i) V = {P, Q, R, S, T, U}

24 U R (ii) E = {(P, P), (P, Q), (P, R), (Q, R), (R, S),
P 20 32 18
(S, T), (S, T)}

(iii) 14

S (c) (i) V = {P, Q, R, S, T}
T 30
(ii) E = {(P, Q), (R, Q), (S, R), (P, S), (S, P),
(b) Total weight = 24 + 20 + 32 + 18 + 30 = 124
(S, T), (T, T)}

(iii) 14 Q P
(b)
Self Practice 5.1d 2. (a)

1. (a) M 8.4 K P Q T
8 3.5 5.2 S RR
KS
11 S 5.9 T

15 4.9 U S
10 34.6 km Kuala Krau

CJ T
3. (a)

(b) K 21.1 km 9.3 km Temerloh

KS M 5.2 21.9 km Lanchang Mentakab
8 3.5 T
30.2 km
S
Karak Bandar Bera

4.9 4.9 km
10
Teriang

CJ (b) Yes, because every pair of vertices is connected

(c) 31.6 km Mervin by one edge. Vertex = 7, Edge = 6
2. (a) rCicheicken Ain
Nleamsai k 4. Route A C D E because it is a safer route even

though Lani had to cycle 300 m more.

Raj 5. (a) (i) P Q R S (ii) P S
Wong
Nurul (b) Route P Q S because I can save RM35 and
Puspa
Atiqah Helen the difference in time is only 9 minutes compared
Faruk
to route P S.
nForioeddles Fried rice

Julia 6. 11 = x1 + x2, x4 = x3 + 11, x2 + x3 = 20,
x1 + 10 = x5, x5 + 10 = x4, x1 = 5,
(b) Types of food. Each type of food is favoured by x2 = 6, x3 = 14, x4 = 25.
more than two pupils.

300

7. (a) D F 3. y = 4x – 5 (3, 7) 4. y = –3x + 4 (1, 1)
C E
A y Ͼ 4x – 5 (2, 4), (–2, 0) y Ͼ –3x + 4 (–1, 8), (–0.5, 7)
Y
B y Ͻ 4x – 5 (0, –6), (4, 5) y Ͻ –3x + 4 (–2, 3), (0, 1)

(b) 3.08 km Self Practice 6.1c
8. (a) X

1. (a) y (b) y

Z

y = —31 x + 3

O x 3
–2 y = –2
(b) (i) {C, R, K, I} y Ͻ –2 y Ͻ —13 x + 3

(ii) {P, G, D, C, R, K, I} –3 O x
3
(iii){E}

9. (b) (i) RM1 080 (ii) 40 (c) y
(c) 13 068
10. (a) third (b) 484 x=2 (d) y y=x+2

CHAPTER 6 Linear Inequalities in yϾx+2 2
Two Variables
O 2 x –2 O x
xഛ2

Self Practice 6.1a (e) y (f) y
3
1. (a) 25x + 45y ഛ 250 or 5x + 9y ഛ 50 y ജ – —21 x – 2 yജx y=x
(b) 2x + 1.5y ഛ 500 or 4x + 3y ഛ 1 000 –3 O x
(c) 0.3x + 0.4y ഛ 50 or 3x + 4y ഛ 500
(d) 1.5x + 3.5y ജ 120 or 3x + 7y ജ 240 –3 3

Self Practice 6.1b – 4 –2 O x
–2
1. Region y Ͼ —32 x – 2
y = – —12 x – 2
y y = —32 x – 2
1 (3, 1)

x y = –23x – 2 (1.5, –1) 2. (a) y (x = 0) (b) y
y Ͼ –32x – 2 (3, 1), (1, –1)
O1 23 y Ͻ –23x – 2 (2, –2), (3, –2) x y Ͼ —21 x y = —12 x
–1 (1, –1) 1 x
(1.5, –1) xഛ0 O
–2 O 2
(2, –2) (3, –2)

Region y Ͻ —32 x – 2

2. Region y Ͼ – –21x + 2 (c) y (d) y
2y = x + 4
y
2
6 (4, 5) x + y ജ –3
(–3, 5) Ox
–3 O x –4
4

2 (2, 1) –3 x + y = –3 2y < x + 4
(–3, 1)
24 x y = – –12x + 2 (e) y (f) y
–4 –2 O (1, –2) 6 Region y Ͻ – –21x + 2
–2
2y + x = 2

y = – –21x + 2 (2, 1) 2 x 1 2y + x ജ 2
y Ͼ – –12x + 2 (–3, 5),(4, 5) O2
y Ͻ – –12x + 2 (–3, 1), (1, –2) O2 x

y ഛ –x + 2 y = –x + 2 Saiz sebenar

301