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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?