Handout 21
COMP 163: Discrete Structures, Spring 2018
Nathan Lopez
Loyola University, Computer Science Department
Problem Set #7
Issued April 9, 2017
Due at 2:00pm on April 20, 2017
1. Consider the following graph with V = {A, B, C, D, E, F } and E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
A
F
2
1
3
4
B
5
6
D
8
9
7
C
(a)
(b)
(c)
(d)
(1
(1
(1
(1
point)
point)
point)
point)
E
10
11
Find all pairs of parallel edges.
Find all loops.
Find all isolated vertices.
What vertices are incident on edge 4?
2. (a) (4 points) Draw the complete bipartite graph K2,5 .
(b) (3 points) How many edges are in K2,5 ?
(c) (5 points) For arbitrary integers m, n 1, how many edges are in the complete
bipartite graph Km,n ?
3. Consider the following graph G:
T
U
Y
W
V
Z
X
1
(a) (7 points) Find the degree of each vertex in G.
(b) (5 points) Does G has an Euler cycle? Why or why not?
4. Consider the following graph:
A
B
C
(a) (5 points) Find the adjacency matrix M .
(b) (3 points) Calculate M 3 (you do not have to show your work for this, you may
use a calculator or other program to compute this matrix multiplication).
(c) (3 points) Use M 3 to find the number of paths of length 3 from A to B.
5. Consider the following tree with root a:
a
c
b
e
d
f
(a) (3 points) Find the level of each node.
(b) (3 points) Find the height of the tree.
2
6. (5 points) Use any algorithm to find a spanning tree of the following graph. Designate
the root of your spanning tree. You do not have to show your work.
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U
Y
W
V
Z
X
Total: 50 points
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Issued April 9, 2017
Due at 2:00pm on April 20, 2017
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1. Consider the following graph with V = {A, B, C, D, E, F} and E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}.
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(a) (1 point) Find all pairs of parallel edges.
(b) (1 point) Find all loops.
(C) (1 point) Find all isolated vertices.
(d) (1 point) What vertices are incident on edge 4?
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2. (a) (4 points) Draw the complete bipartite graph K 2,5.
(b) (3 points) How many edges are in K2,5?
(c) (5 points) For arbitrary integers m, n > 1, how many edges are in the complete
bipartite graph Km,n?
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a homework8solutions.pdf (page 5 of 5)
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4. Consider the following graph:
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(a) (5 points) Find the adjacency matrix M.
(b) (3 points) Calculate M: (you do not have to show your work for this, you may
use a calculator or other program to compute this matrix multiplication).
(c) (3 points) Use M3 to find the number of paths of length 3 from A to B.
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5. Consider the following tree with root a:
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(b) (3 points) Calculate M? (you do not have to show your work for this, you may
use a calculator or other program to compute this matrix multiplication).
(c) (3 points) Use M to find the number of paths of length 3 from A to B.
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(a) (3 points) Find the level of each de.
(b) (3 points) Find the height of the tree.
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a homework8solutions.pdf (page 5 of 5)
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h_1_.pdf (page 3 of 3)
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6. (5 points) Use any algorithm to find a spanning tree of the following graph. Designate
the root of your spanning tree. You do not have to show your work.
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Total: 50 points
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Irk4soluti
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a homework8solutions.pdf (page 5 of 5)
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