Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 1
Due: Thursday, September 23 by 4:00 in Assignment box 37
1. Let f (x) = 3x3 2x + 1. Trace the running of Horners algorithm to nd f (2).
2. Describe an algorithm that, upon input of integers a and b and a natural number n,
out
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 4
Due: Tuesday, November 9 by 4:00 in Assignment box 37
1. Let D be a digraph with adjacency matrix A. What does the sum of the entries in row i of A represent?
2. Consider the following digraph D.
v1
v5
v2
v3
v4
(a) Find its a
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240 Assignment 3 solutions
1. (a) The degree of each vertex can be found by counting the number of 1s in each row
of the adjacency matrix. We see that each vertex has degree at least 5. Since there
are 10 vertices, we conclude by Diracs theor
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 1 Solutions
1. We have that a0 = 1, a1 = 2, a2 = 0, a3 = 3 and x = 2.
i
1
2
3
S
3
a2 + 3x = 6
a1 + 6x = 10
a0 + 2x = 21
So f (2) = 21.
2. Set S =
for i = 0 to n 1 do
if ai b 0 (mod n), then replace S by S cfw_i
(could also hav
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
Memorial University of Newfoundland
Department of Mathematics and Statistics
Pure / Applied Mathematics 3240
Applied Graph Theory
Fall 2010
Instructor:
Oce:
Email:
Phone:
Dr. Andrea Burgess
HH2016
aburgess@mun.ca
7374321
Lectures:
Oce Hours:
Tuesdays an
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 6
Not to be handed in.
1. For each i cfw_1, 2, 3, 4, 5, nd the weight of a minimum spanning tree in G cfw_vi . What can you
conclude about the weight of a leastweight Hamiltonian cycle in G?
v1
4
8
11
11
v5
v2
8
4
11
7
v4
7
8
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 4 Solutions
1. The (i, j)entry of A is 1 if and only if (vi , vj ) is an arc. So the number of 1s in the ith row
of A is the number of arcs of the form (vi , vj ), i.e. the outdegree of vi .
2. (a)
A=
0
1
0
0
1
1
0
1
1
0
0
1
0
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 2
Due: Thursday, October 7 by 4:00 in Assignment box 37
1. (a) Does there exist a graph with the given degree sequence? If so, draw such a graph. If not, explain
why not.
(i) (7, 6, 3, 2, 2, 1)
(ii) (5, 3, 3, 3, 3, 3, 0)
(iii)
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 2 Solutions
1. (a)
i. No. There cannot be an odd number of odd vertices.
ii. Yes. Such a graph is drawn below. (Note the isolated vertex.)
iii. No, there cannot exist such a graph. Such a graph would have to have one vertex (of
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 3
Due: Thursday, October 28 by 4:00 in Assignment box 37
1. (a) Let G be a graph with the following adjacency matrix. Without drawing the graph, show that G
is Hamiltonian. (You do not need to give a Hamiltonian cycle, only to
Introductory Vector Algebra and Discrete Mathematics
MATH 1340

Summer 2007
PMAT / AMAT 3240
Assignment 5
Due: Tuesday, November 23 by 4:00 in Assignment box 37
1. (a) Suppose that in the tree T , 100 vertices have degree 1 and 20 vertices have degree 6. Of the
remaining vertices, half have degree 2 and half have degree 4. How ma