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
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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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 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
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 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: 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/am3i
Linear Algebra
2006-2007
6. Eigenvalues and eigenvectors
In this section we will study properties of linear transformations of a nite-dimensional
vector space V .
Any function or property of a square matrix which is not altered under a chan
MAT-2-mi3/am3i
Linear Algebra
2006-2007
5. Polynomials
We take a break from the study of linear maps in vector spaces to discuss polynomials. We shall return to linear maps again at the end of this section.
5.1. Polynomial multiplication. In Example 1.8 w
MAT-2-mi3/am3i
Linear Algebra
2006-2007
4. Matrices
In this section we will learn how to translate questions about linear maps into questions about matrices, which we will then be able to answer by applying the computational techniques at our disposal and
MAT-2-mi3/am3i
Linear Algebra
2006-2007
3. Linear maps
In the previous section we have learnt about vector spaces by studying objects
(subspaces, bases,.) living in a xed vector space. In this section we will learn
how to relate dierent vector spaces. Thi