Cenartech sht nj kompani inxhinierike e SHBA, me rreth 400 punonjs. Firma bn pajisjet pr
monitorimin e proceseve pr kompanit prodhuese t ushqimit dhe industrit farmaceutike dhe
kozmetike. Konsumatort
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
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 h
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 t
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
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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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) w
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)
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 .
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 adja
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 verte
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 (P
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.
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
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
algorit
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 (Pr
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
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 (Ha
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-emp
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 (Ha
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
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
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
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 (Pre
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
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
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 theo
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 proper
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 se
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 ap