The document discusses different concepts in graph theory including: - Euler paths and circuits which travel along each edge of a graph once. An Euler circuit is a closed path. - Hamiltonian paths and circuits which pass through each vertex of a graph once. - Weighted graphs where weights are assigned to edges to represent attributes like distance or cost. - Networks which are weighted graphs used to model real-world systems like transportation or computing. - Critical paths which identify the minimum time needed to complete interconnected tasks in a network.

1. CST 5 Math
A-maze-ing
Math
Graphs

2. The Königsberg Conundrum
 In the old city of Konigsberg there
used to be only 5 bridges.
 People could take a round trip of all
the bridges by crossing them only
once.
 Go to page 70 in the text and trace
with your finger the path you would
take.

3. The Seven Bridges
 Then two more bridges were built.
 People tried but could not do a round
trip and cross each bridge only once.
 Try this using the picture on the next
slide.

4. Leonard Euler
 This smart guy, Leonard Euler, (pr.
Oiler) was able to show why a round
trip was impossible.
 He used dots to represent the land
and lines to represent the bridges.

5. The Oiler does it!
 So Lenny showed that no matter
where you started, you could not help
but pass over a bridge two times.
 By doing this he introduced graph
theory which shows how the elements
of a set relate to each other.

6. First, Some Definitions
 graph
 Informally, a graph is a finite set of dots
called vertices (or nodes) connected by
links called edges (or arcs).

7. Definitions
 adjacent
 Two vertices are adjacent if they are
connected by an edge.

8. Definitions
 Degree: The degree (or valence) of
a vertex is the number of edge ends
at that vertex.
 For example, in this graph all of the
vertices have degree two.

9. Activities
 Book 2, p.18, Q. 1-7, some orally
 P. 19, Q. 9, 10, 14, 15

10. More Definitions
 complete graph
 A complete graph with n vertices is a
graph with n vertices in which each
vertex is connected to each of the others
(with one edge between each pair of
vertices).
 Here are the first five complete graphs:

11. Definitions
 connected
 A graph is connected if there is a path
connecting every pair of vertices.


12. Activities
 P. 79, Q. 1-7
 P. 81, Q. 8-11

13. Labyrinths
 So, the island in Königsberg and the
bridges created a maze or a labyrinth.
 Labyrinths have existed for thousands of
years.
 According to legend, King Minos created a
labyrinth on the island of Crete.
 At the centre was his son, a half-man half-
bull called a Minotaur (a/k/a Bob).
 If you could get out of the labyrinth before
Bob got you, you survived.

14. Bob and His Maze

15. Bob Dies…
 Theseus kills Bob.
 He then escapes.
 He goes home.

16. Okay
 So that was not actually the maze.
 What did your graph look like?

17. Activities
 On page 85, do Q. 1-4 orally.
 On page 86, do Q. 6-10
 On Page 87, do Q. 13, 16, 17, 21

18. Labyrinth Project
 You and a partner will pick a maze.
 Draw the labyrinth.
 Draw a graph of the labyrinth
showing the vertices and the edges.
 This is a C1 task – math
communication – constructs and uses
networks of concepts;
 You will write a short paragraph
about the labyrinth, how it works,
and what you have learned.

19. Chartres Cathedral Prayer Maze

20. Brain Maze

21. Theseus Maze

22. Circular Maze

23. Euler Paths and Circuits
 Euler path: A path that travels over
each edge once and only once in a
connected graph.
 Euler Circuit:
 An Euler path
 that is closed.

24. Special Case
 An Euler path or circuit only exists if a
graph has EXACTLY 2 vertices whose
degrees are odd numbers.
 An Euler path exists when the
degrees of all the vertices are even
numbers.

25. Euler Circuit:
Degree of Vertices

26. Drawing Euler Paths
 An Euler path must start at a vertex
having an odd-numbered degree and
end at another vertex with an odd-
numbered degree.
 An Euler circuit can begin at any
vertex and ends at the same vertex.

27. Hamiltonian Paths
 Hamiltonian Path: A path that passes
through every vertex once and only
once.
 Hamiltonian Circuit: A Hamiltonian
path that finishes at the same vertex.

28. Distance of a Path
 D(P,Q) is the shortest distance
between two points.
 Each edge/line is considered to have
length one.

29. Activities
 Page 27, Q. 1-7

30. Tree Diagrams
 A tree diagram is a
connected graph
without a simple
circuit.
 This can be used in
planning jobs,
electrical circuits
and plumbing.
A
B C D

31. Directed Graph or Digraphs
 A directed graph is a graph in which
each edge has a direction – called an
arc;
 The arc has only one direction that
can be followed; E.g one way street
 A path or circuit is SIMPLE if it
contains no repeat arcs;

32. Weighted Graph
 A weighted graph is a graph,
directed or not, in which a weight is
attributed to each edge;
 The weight of a path is the sum of
the weights of the edges that make
up the path;

33. Activities
 Page 37, Q. 1, 3, 4
 Page 41, Q. 13, 15, 16

34. Value of a Path
 The value of a path is the total of its
weights.
 It can be a maximum or a minimum.
 To find the minimum, start with the
path of lowest weight, and add
additional edges until all the nodes
are connected.

35. Networks
 A network is a graph in which every
edge is assigned a weight.
 A weight can indicate time or cost.
 The weight of a path corresponds to
the sum of the weights of the edges
that make up the path.
 A network can be directed or
undirected.

36. Chromatic Number
 The chromatic number is the
minimum number of colours
necessary to colour all of a graph’s
vertices without any 2 adjacent
vertices being of the same colour.
 It is also applied to maps.

37. Critical Path
 The critical path corresponds to a
simple path of maximum value.
 Critical paths are used to determine
the minimum amount of time required
to carry out a task comprising several
steps.
 To do this, you must know which steps
are pre-requisites (needed ahead of
time) for other steps, and which steps
can be carried out at the same time.

38. Activity
 Page 52, Q. 1,2,5
 Page 54, Q. 9, 10, 12, 13, 19