MAT 226 Discrete Mathematics
Homework Assignment 10 Solutions
1.
Construct adjacency matrices for the graphs below.
0
1

0

0
0

1
0
0

0

0
0
1
0
1
1
0
1
0
1
0
1
0
0
1
0
1
0
0
0
1
1
0
1
1
0
1
0
0
0
0
0
0
1
0
1
0
0
1
0
1
1
1 
0

1
1

0
2
0 
1

0
0 
2.
Construct the graphs corresponding to the adjacency matrices below. A is an
undirected graph; B must be a directed graph.
1
0

A  1

0
2
0
0
0
2
1
1
0
0
1
1
0
2
1
0
1
2
1 
1

1
0 
0
0

B  1

1
1
1
0
0
0
1
1
1
0
1
0
0
1
1
0
0
1
1 
1

1
0 
Graph A
Graph B
3.
Determine whether the pair of graphs is isomorphic and justify your conclusion.
The justification in case the graphs are not isomorphic should identify the invariant
that is not preserved (e.g., number of edges, degrees of vertices, etc.). In case the
graphs are isomorphic, identify an isomorphism and, using the corresponding
orders, show that the adjacency matrices are the same.
The graphs are not isomorphic. In the left graph, the two vertices of degree 3 are
adjacent; they are not adjacent in the right graph.
4.
Repeat #3 for the following problems from the text.
a) Section 10.3, Problem 36
The graphs are not isomorphic. One has a vertex of degree four and the other
does not.
b) Section 10.3, Problem 38
The graphs are isomorphic. One such isomorphism is
u1  v1; u2  v 3 ; u3  v 2 ; u4  v 5 ; u5  v 4
c) Section 10.3. Problem 40
The graphs are not isomorphic. One has a vertex of degree four and the other
does not.
5.
Section 10.4, Problem 6
#3 Three connected components
#4 One connected component (the entire graph)
#5 Two connected components
6.
Section 10.4, Problem 12
#12a Weakly, but not strongly connected
#12b Both weakly and strongly connected
#12c Neither weakly nor strongly connected
7.
Section 10.4, Problem 14a
The strongly connected components are the subgraphs with vertex sets {a, b, e},
{c}, {d}
8.
Section 10.4, Problem 20
The graphs are not isomorphic. One way to see this is that G has a simple circuit
containing only degree three vertices u1u5u6u2u1  but H does not have such a
simply circuit. Another way is to notice in H there is a degree two vertices
connected to two degree three vertices; that does not happen in G.
9.
Section 10.4, Problem 26abc
0
1
0
A=
1
1
(0
4
9
A3 = 4
7
7
(5
1
0
1
0
1
1
9
6
9
4
10
7
0
1
0
1
0
1
1
0
1
0
1
0
4
9
2
8
4
7
1
1
0
1
0
1
7
4
8
2
9
4
0
1
1
0
1
0)
7
10
4
9
6
9
3
1
2
A2 =
1
2
(2
5
7
7
4
9
4)
23
20
A4 = 21
15
25
(20
1
4
1
3
2
2
2
1
3
0
3
1
20
35
17
28
26
25
a) There are no paths of length 2. [The (3, 4) entry of A2]
b) There are 8 paths of length 3. [The (3, 4) entry of A3]
c) There are 10 paths of length 4. [The (3, 4) entry of A4]
1
3
0
3
1
2
21
17
24
10
28
15
2
2
3
1
4
1
15
28
10
24
17
21
2
2
1
2
1
3)
25
26
28
17
35
20
20
25
15
21
20
23)
10. Section 10.4, Problem 32
Cut vertices: c and d
Bridges: {c, d}
11. a) What are the bridges (cut edges) of the graph in Section 10.4, Problem 4?
Every edge is a bridge
b) What are the cut vertices and bridges of the graph in Section 10.4, Problem
50b?
Cut vertices: c
Bridges: None
c) What are the bridges and bridges of the graph in Section 10.3, Problem 13?
Cut vertices: c
Bridges: {c, d}