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 colouring problem Map colouring problem Museum guard problem Museum guard problem Königsberg bridge problem Königsberg bridge problem Travelling salesman problem Chinese postman problem Chinese postman problem

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.

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 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 A B C D Can 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 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.

Can you find Hamilton Circuits for the vertices of other 3D shapes?