F4 Chapter 5 Network in Graph Theory

Vertex ● ● 线/边

黑点/顶点 Edge

●

网络

黑点/顶点
线/边

社交网络

Chapter 5 Network

5.1 Network

Example: C E
D
A

Vertex B

V = {A, B, C, D, E} n(V) = 5
Edge n(E) = 6

E = {(A, B), (A, C), (B, C), (B, D), (C, D), (C, E)}

5. A graph is said to be connected if there is a path between each pair of vertices.

6. A graph is a graph which has at least a pair of related vertices. (at least a pair
of vertices is connected by an edge)

E ●P

AD S● ● ● ●T

F●

G R● ●Q

BC ● disconnected
graph
U

7. A simple graph has no loops and no multiple edges.
8. The degree, d , is the number of edges that connect two vertices.
9. The sum of degrees of the graph is twice the number of edges.

Number of vertices = 6
Number of edges = 7
Sum of degrees = 2(7)

= 14

Simple graph

Not simple graph

With arrows With multiple edges With loop

10. An edge that starts and ends at the same vertex is known as a loop.

V = {P, Q, R}
Q n(V) = 3

E = {(P, Q), (P, R), (Q, R), (Q, Q)}
P R n(E) = 4

Sum of degrees = 8

A BC V = {A, B, C, D, E, F}
FE n(V) = 6

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

n(E) = 10 Degree of vertex A = 3
Sum of degrees = 20 Degree of vertex B = 3
Degree of vertex C = 3
Degree of vertex D = 4
Degree of vertex E = 3
Degree of vertex F = 4

Draw a simple graph

V = {1, 2, 3, 4, 5}

E = {(1, 2), (1, 4), (1, 5), (2, 3), (2, 4), (3, 4)}

●1 ●2 ●3 1●

●5

● ● 2● ●
3●
5 4 4

Draw a graph with multiple edges and loops

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

●A ●B C●

D●

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

(a) 3, 1, 2, 1, 2 (b) 2, 3, 2, 1, 1, 3

(a) Sum of degrees (b) Sum of degrees
=3+1+2+1+2 =2+3+2+1+1+3
=9 = 12

The graph cannot be drawn The graph can be drawn because
because the sum of degrees is odd. the sum of degrees is even.

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3. (a)
(b)

4. (a)
(b)

Directed graph and undirected graph
1. A directed graph is a graph which each of its edges is assigned a

directed shown by arrows. The movement from one vertex to another
vertex is only one directed.

B

A Vertex
D
A

CB

C

D

2. An undirected graph is a graph which its edges are not assigned any
direction. It has two-way direction.
B

AD C

Vertex V 2 13
A 1 23
B 2 35
C 1 01
D 6 6 12

Total

5. A weighted graph is a graph such that a number is allocated to each
of its edges.

6. These weights represent cost, distance, time and so on.
7. A weighted graph can be an undirected graph or a directed graph.

Example:

Graph A Graph B

Graph D

Graph C

Graph A Graph B
Graph D
Graph C

Identify:
(a) the directed graphs A, B, D
(b) the weighted graphs C, D
(c) the unweighted graphs A, B
(d) the directed and weighted graphs D
(e) the undirected and weighted graphs C

Example:

Draw a directed graph based on the following information:
V = {P, Q, R, S}
E = {(P, Q), (S, P), (S, Q), (Q, S), (Q, R). (S, R), (R, R)}

Vertex V 1 1
P 2 2
Q 3 1
R 1 3
S 7 7

Total

Example:

The diagram shows an incomplete directed graph.

Example:
The diagram below shows one way paths that Mohan can choose for his running practice.

Given that vertex P is the starting position and vertex S is the ending position before he goes
home. Determine

(a)The shortest distance from P to S.
(b)The longest distance from P to S.
(c)The vertices that must he passed through if the distance of the one-way run is between

0.8 km and 1.0 km.

Solution:
(a)Shortest distance = P R  S

= 500 m + 300 m
= 800 m

(b) Longest distance = P  Q  R U  T  S
= 400 m + 300 m + 700 m + 300 m + 300 m
= 2000 m

(c) P  U  T  S = 300 m + 300 m + 300 m
= 900 m

Example:
The diagram shows one-way paths that Kumari can choose for her running
practice. Given that the distance from A to D go through B and C is equal to the
distance from A to D go through E. Find the value of x in km.

Solution :
x + 0.4 km = 0.45 km + 0.30 km + 0.50 km

x = 1.25 km – 0.4 km
= 0.85km

Example:

Subgraphs and Trees

A subgraph is part of a graph or the whole graph redrawn without changing the
original positions of the vertices and edges.

Page 142

Yes Yes Yes Yes No

No No Yes No No Yes

no loops
no multiple edges

All vertices are connected by
only one edge

vertices (n) > edges (n – 1)





Page 142

No No Yes No
n(V) = 5 n(V) = 8 n(E) = n(V) – 1 n(V) = 5

edges = 4 edges = 5