1
The adjacency matrix of a graph
There are discussions of the adjacency matrix of a graph in MR, pp. 214224, in Epp, 2nd Ed,
pp. 640654. Both of these sections are followed by examples sections with many examples
for you to do. Also, you may recall from

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 6
Questions handed in on Friday, 5th November
6.1 (Prepare for discussion at the tutorial) Without drawing the graph, use Kruskals
algorithm to construct a spanning tree for the complete graph

mi3, Semester 1, 2010/11
Counting: Example Sheet 6
Hand in the questions indicated BEFORE the lecture on Friday,
5th November
Remember that credit will be deducted if work is poorly presented.
6.1 (Prepare for discussion at the tutorial) Without drawing t

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 5
Questions handed in on Friday, 29th October
5.1 (5 points) Let A be the set of all functions
f : cfw_1, 2, 3, 4, 5 cfw_1, 2, 3.
(i) How many elements are there in A?
(ii) Write down two diere

mi3, Semester 1, 2010/11
Counting: Example Sheet 5
Hand in the questions indicated BEFORE the lecture on Friday,
29th October
Remember that credit will be deducted if work is poorly presented.
5.1 (Hand-in) Let A be the set of all functions
f : cfw_1, 2,

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 4
Questions handed in on Friday, 22nd October
4.1 (2 points) (Hand-in)
(a) Count the number of partitions of cfw_1, 2, 3, 4 into 2 non-empty subsets
(using the formula). Describe them explicitl

mi3, Semester 1, 2010/11
Counting: Example Sheet 4
Hand in the questions indicated BEFORE the lecture
on Friday, 22nd October
Remember that credit will be deducted if work is poorly presented.
4.1 (Hand-in)
(a) Count the number of partitions of cfw_1, 2,

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 3
Questions handed in on Friday, 15th October
Note that this time not all of the handed-in questions will be marked!
3.1 (2 points) Count the number of surjective functions
f : cfw_1, 2, 3, 4,

mi3, Semester 1, 2010/11
Counting: Example Sheet 3
Hand in the questions indicated BEFORE the lecture
on Friday, 15th October
Note that this time not all of the handed-in questions will be marked!
3.1 (Hand-in) Count the number of surjective functions
f :

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 2
Questions handed in on Friday, 8th October
2.1 (Prepare for discussion at the tutorial) Let A be a nite set with
#A = n and let P (A) be the power set of A. Show that #P (A) = 2n .
Deduce tha

mi3, Semester 1, 2010/11
Counting: Example Sheet 2
Hand in the questions indicated BEFORE the lecture
on Friday, 8th October
Remember that credit will be deducted if work is poorly presented.
1.1 (Prepare for discussion at the tutorial) Let A be a nite se

mi3, Semester 1, 2010/11
Counting: Example Sheet 8
Hand in the questions indicated BEFORE the lecture on Friday,
19th November
Remember that credit will be deducted if work is poorly presented.
Useful denitions:
A planar graph is a graph that can be draw

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 7
Questions handed in on Friday, 12th November
7.1 (Hand-in: 3 marks) The following graph represents 5 cities and the roads between
them. The weight of each edge corresponds to the distance bet

0
Halls Marriage Theorem
The material in this section is discussed in Wilsons book Introduction to graph theory,
Chapter 8, pp 115-118.
The Marriage problem is the following problem.
Suppose that we have a nite set of boys, each of whom knows several girl

0
Planar graphs
The material in the planar graphs section is covered in Wilson and Watkins, Chapter
11, pp 215-226, except for Halls Marriage Theorem, which is discussed in Wilsons book
Introduction to graph theory, Chapter 8, pp 115-118.
The planar graph

0
mi3 Graph Theory
1
Trees
1.1
Spanning tree algorithm
The algorithm for constructing a spanning tree which is considered in this subsection is particularly elegant.
It simply examines each edge in a graph, deciding whether it should be in the spanning tr

Trees
A graph G is a tree if G is connected and has no cycles.
Given a graph G, a spanning tree of G is a subgraph which includes
all vertices of G and is a tree.
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()*+
/.-,
()*+
/.-,
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3
4
5

0
1
mi3 Graph Theory
Introduction
Graph theory started with Eulers solution in 1736 of the Knigsberg
o
bridge problem.
The city of Knigsberg in East Prussia (now the city of Kaliningrad in
o
Russia) was built at the conuence of the North and South Pregel

Equivalence Relations
Recall that any function f : A B determines a partition
of A.
Example f : cfw_0, 1, . . . , 11 cfw_0, 1, 2 dened by f (k ) :=
remainder when k is divided by 3.
Set
A0 := f 1 (0) = cfw_
A1 := f 1 (1) = cfw_
A2 := f 1 (2) = cfw_
The

Mathematics for Informatics 3
Counting
Set Theoretic language
Sets can be arbitrary, but we will mainly get interesting results for nite
sets.
Membership a X means that a is an element of the set X . We often
say: a is in X .
Subsets Given sets A and B ,

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 9
Questions handed in on Friday, 26th November
9.1 (Prepare for discussion at the tutorial)
(i) Draw the directed graph which has adjacency matrix
A(
(ii) Use the adjacency matrix to cal

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 8
Questions handed in on Friday, 19th November
8.1 (Hand-in: 4 marks) Is there a connected planar (simple) graph such that:
(a) each vertex has degree 4?
(b) each vertex has degree 6?
Solution:

mi3, Semester 1, 2010/11
Counting: Example Sheet 9
Hand in the questions indicated BEFORE the lecture on Friday,
26th November
Remember that credit will be deducted if work is poorly presented.
9.1 (Prepare for discussion at the tutorial)
(i) Draw the dir

mi3, Semester 1, 2010/11
Counting: Solutions to Example Sheet 1
Questions handed in on Friday, 1st October
1.1 (2 points) Let A, B and C be sets. Show that:
(i) (A B ) C = (A C ) (B C );
(ii) (A B ) C = (A C ) (B C ).
Solution:
(i) Let x (A B ) C . Then x

mi3, Semester 1, 2010/11
Counting: Example Sheet 1
Hand in the questions indicated BEFORE the lecture
on Friday, 1st October
Remember that credit will be deducted if work is poorly presented.
1.1 Let A, B and C be sets. Show that:
(i) (A B ) C = (A C ) (B

MAT-2-mi3/am3i
Linear Algebra
2006-2007
7. Linear algebraic codes
Algebraic coding theory is the branch of mathematics which deals with the problem
of communicating across a noisy channel. Coding theory is usually told as a story
of two characters, say, A

MAT-2-mi3 Linear Algebra 2010
Tutorial Sheet 6
Due on Friday 5 November
Credit will be deducted for poorly presented work.
Problem 6.1. (12 marks) For each of the following systems of linear equations:
a.
x 2y + 3z = 4
2x 3y + 4z = 3
3y + 2z = 1
b.
7x1 +

MAT-2-mi3 Linear Algebra 2010
Tutorial Sheet 6
Due on Friday 5 November
Credit will be deducted for poorly presented work.
Problem 6.1. (12 marks) For each of the following systems of linear equations:
a.
x 2y + 3z = 4
2x 3y + 4z = 3
3y + 2z = 1
b.
7x1 +

MAT-2-mi3 Linear Algebra 2010
Tutorial Sheet 5
Due on Friday 29 October
Credit will be deducted for poorly presented work.
Problem 5.1. (6 marks) Let A : R4
3 4 matrix
1 2
0 2
3 4
R3 be the linear map dened by the
12
2 0 .
16
Find a basis for its kernel

MAT-2-mi3 Linear Algebra 2010
Tutorial Sheet 5
Due on Friday 29 October
Credit will be deducted for poorly presented work.
Problem 5.1. (6 marks) Let A : R4
3 4 matrix
1 2
0 2
3 4
R3 be the linear map dened by the
12
2 0 .
16
Find a basis for its kernel