Graph Theory
What is Graph Theory?
This is the study of structures called ‘graphs’.
These graphs are simply a collection of points
called ‘vertices’ (or ‘nodes’) connected by ‘edges’
(or ‘arcs’).
edge
vertex
Why is it useful?
Real life problems and problems from other areas
of mathematics can be turned into Graph Theory
problems.
Chinese
Shortest
path
problem
postman
Travelling
problem
salesman
Museum guard
problem
problem
Optimising
Königsberg
computer
Map colouring
bridge
networks
problem
problem
The theorems and knowledge about Graph Theory
can then help us solve these problems.
Six Degrees of Kevin Bacon
The ‘Shortest Path Problem’
applied to the Six Degrees of Kevin
Bacon. Actors are represented as
vertices. If two actors are in the
same film or TV show they are
connected by an edge. An actor’s
‘Bacon Number’ is the degrees of
separation he is from Kevin Bacon.
In other words, the fewest edges
that must be travelled to get to
the Kevin Bacon vertex.
Six
Degrees
of
Kevin
Bacon
Christopher
Plummer
2
Kevin
Elvis was in
Kevin Bacon
Bacon
Change
of
Of course all these
in index
has an
0 actors were
Habit with
films with many other actors, so of
the
0 Tom
Asner
but
was
graph is much larger.
Hanks
never in a film
Edward
1
with KB so has
Asner
Edward Asner was
an index of 2
1
in the film JFK
Elvis
Gary
with KB so has an
Jordan
Presley
Sinese
index of 1
Nagai
2
1
2
Leonhard Euler
• 1707-1783
• Swiss
• Contributed to many
areas of maths:
– Optics
– Graph theory
•
•
•
•
Great at mental maths
Photographic memory
Devout Christian
e iπ = -1
Königsberg Bridges
Historical problem ‘solved’ by Euler in 1735.
A
C
B
D
Can you walk around the city
crossing each bridge exactly once?
The city can be
represented as
a graph.
Start at one vertex and see if
you can ‘walk’ over all the
edges exactly once.
What happens if you
remove an edge?
A
C
B
D
Does it matter which edge
you remove?
Why are some bridge
problems solvable and
some not?
William Rowan Hamilton
• 1805-1865
• Irish
• Contributed to many
areas of maths:
–
–
–
–
Optics
Mechanics
Graph theory
Algebra
• Great linguist
A Hamiltonian cycle or circuit is a path that takes
you through every vertex exactly once and finish
where you started.
Hamilton invented a mathematical game in 1857
using a dodecahedron.
You had to find a path around
the edges so that you visit
each vertex once and only
once.
These is easier to manage in 2-dimensions if we
draw the dodecahedron as below.
Can you find Hamilton
Circuits for the
vertices of other 3D
shapes?