Math 3373 Assignment 1 (Solutions)
Math 3373 Assignment 1 (Solutions)
1.2 The vertex set is
V  1, 2, 3, 4, 5, 6, 7, 8
and the edge set is
E  12, 23, 23, 23, 34, 14, 56, 67, 57, 57, 55, 88, 88.
1.3 Answer varies.
1.6 The two graphs are isomorphic.
1
2
1
3
5
7
4
6
3
8
4
7
6
5
8
2
1.9 29 & 23.
1.10 (i) Since vertices are labelled, every pair of distinct vertices can be a possible edge
in the simple graph. From n vertices, there are
n  nn  1
nC2 
2
2
possible edges. For each possible edge, it may or may not be belong to the edge
set. So the number of distinct graphs are
2
nn1
2
.
(ii) Among them, the number of graphs with exactly m edges is
nn1
2
1.12
1 of 7
Cm 
nn1
2
m

nn1
2
nn1
m
2
!
!m!
.
Math 3373 Assignment 1 (Solutions)
Degree sequence: 1, 2, 2, 3
Two vertices have odd number degrees.
Degree sequence: 1, 2, 2, 3, 4
Two vertices have odd number degrees.
Degree sequence: 2, 3, 3, 4, 4
Two vertices have odd number degrees.
1.14 (i) The number of edges is
m  1 1  2  3  4  5
2
One such graph is
(ii) No. For simple graph with 4 vertices, the maximum possible vertex degree is 3.
1.15 Proof. For a simple graph, every vertex has a degree number from 0 to n  1. If no
two vertices have the same degree number, then the degree sequence of the graph
must be 0, 1, 2, 3, 4, . . . , n  1. The total degree is now
n1
k 
k0
1 nn  1
2
which is an odd number. This contradicts the Hand Shaking Lemma (Theorem 1.1).
1.17
2 of 7
Math 3373 Assignment 1 (Solutions)
u
v
w
e
G:
x
y
u
Ge :
x
z
v
w
y
z
v
w
ux
G\e :
y
z
1.18 If G has n vertices and m edges, v is a vertex with degree k and e is an edge, then
G  e has n vertices and m  1 edges;
G  v has n  1 vertices and m  k edges;
G\e has n  1 vertices and m  1 edges.
1.19 The complements are
3 of 7
Math 3373 Assignment 1 (Solutions)
G
,G 
G
,G 
G
,G 
1.21 For the following graph
The adjacency matrix and the incidence matrix are
A
M
1.22 The graph is
4 of 7
u
0 1 2 0
v
1 4 3 0
w
2 3 0 1
z
0 0 1 0
u
1 1 1 0 0 0 0 0 0
v
1 0
w
0 1 1 1 0 0 1 1 1
z
0 0 0 1 0 0 0 0 0
0 0 2 2 1 1 1
Math 3373 Assignment 1 (Solutions)
2
5
3
1
4
1.23 The graph is
3
2
5
1
4
1.28 (i) the number of edges in K 12 is
m
12
2
 66
(ii) the number of edges in K 6,8 is
m  6  8  48
(iii) the number of edges in Q 5 is
5
m  2  5  80
2
(iv) the number of edges in W 10 is
m  9  9  18
(vi) the number of edges in C 8 is 8, the number of edges of C 8 is
m  8  7  8  20
2
1.31 Answers vary.
1.34
5 of 7
Math 3373 Assignment 1 (Solutions)
K2,2,2
K3,3,2
Number of edges in K 3,4,5 is
m
6 of 7
34  5  43  5  53  4
 47
2