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’).

vertex

edge

Why is it useful?Real life problems and problems from other areas of mathematics can be turned into Graph Theory problems.

Optimising computer networks

Shortest path

problem

The theorems and knowledge about Graph Theory can then help us solve these problems.

Map colouringproblem

Museum guardproblem

Königsberg bridge

problem

Travelling salesman problem

Chinese postmanproblem

Six Degrees of Kevin BaconThe ‘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.

Gary Sinese

1

Six Degrees of Kevin Bacon

Jordan Nagai

2

Tom Hanks

1

Christopher Plummer

2

Elvis Presley

2

Edward Asner

1

Kevin Bacon has an index

of 0

Edward Asner was in the film JFK

with KB so has an index of 1

Kevin Bacon

0

Elvis was in Change of Habit with

Asner but was never in a film with KB so has an index of 2

Of course all these actors were in films with many other actors, so the

graph is much larger.

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 BridgesHistorical problem ‘solved’ by Euler in 1735.

A

B C

DCan you walk around the city

crossing each bridge exactly once?

A

B C

D

What happens if you remove an edge?

Does it matter which edge you remove?

Why are some bridge problems solvable and some not?

The city can be represented as a graph.

Start at one vertex and see if you can ‘walk’ over all the edges exactly once.

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.

You had to find a path around the edges so that you visit each vertex once and only once.

Hamilton invented a mathematical game in 1857 using a dodecahedron.

These is easier to manage in 2-dimensions if we draw the dodecahedron as below.

http://upload.wikimedia.org/wikipedia/commons/6/60/Hamiltonian_path.svg

Can you find Hamilton Circuits for the

vertices of other 3D shapes?