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Graph theory is not...

Graph Theory Summer Math Workshop for High School & Middle School Teachers Sarah Holliday June 12, 2009. Graph theory is not. Cartesian plotting. Graph theory is structure and relationships. Alice and Bob take geometry together Alice and Chuck take French together

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Graph theory is not...

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  1. Graph TheorySummer Math Workshop for High School & Middle School Teachers Sarah HollidayJune 12, 2009

  2. Graph theory is not... • Cartesian plotting

  3. Graph theory is structure and relationships • Alice and Bob take geometry together • Alice and Chuck take French together • Bob and Dave take history together • Chuck and Ed take English together • Bob and Ed take Latin together • Dave and Ed take PE together

  4. Graph theory is structure and relationships A B C E D

  5. Graph theory is not... • Just pictures

  6. Graph theory includes • Colouring • Planarity • Isomorphisms • Counting • Coding • Designs

  7. Graph theory is problem solving • Unlike an algebraic question in which a specific numeric answer is sought, a graph theory problem will more often ask for a sentence. The results from graph theory problems are abstract provide a launching point for writing across the curriculum and group discussions

  8. Colouring: • Colouring a graph is the process of assigning labels to the vertices. • A proper colouring is one in which no two adjacent vertices have the same colour or assigned label. • A minimum proper colouring is a proper colouring using the fewest possible colours.

  9. Colouring:

  10. A colouring: 2 2 1 3 5 2 3 3 4 2

  11. A proper colouring: 2 2 1 3 5 2 5 3 4 2

  12. A minimum proper colouring: 2 2 1 3 1 2 1 3 4 2

  13. A colouring problem • A department chair needs to schedule nine classes A through I in as few rooms as possible. There are certain conflicts that must be avoided. How can the chair schedule the classes?

  14. A colouring problem • Sue teaches classes A, B, C, and D. • Tim teaches classes E, F, and G. • Una teaches classes H and I. • Classes A, F, and I use the projector. • Classes B, E, G, and H use the legos.

  15. A colouring problem A B C I D H E G F

  16. A colouring problem 1 2 A B 3 3 C I 4 D H 4 E 1 G F 3 2

  17. A colouring problem • One solution: • In time slot 1, classes A and E • In time slot 2, classes B and F • In time slot 3, classes C, G and I • In time slot 4, classes D and H

  18. Colouring: maps • We can use a graph to represent physical relationships. We will mark a vertex for each state, and draw an edge between two vertices that share a border (not just a corner).

  19. Colouring: maps

  20. Colouring: maps

  21. Colouring: maps • How many colours are required to properly colour a map or a graph that represents a map? • The second example shows us that four colours suffice. • A graph that has a map representation is called planar.

  22. Planarity: • A graph that can be represented using vertices and edges that do not cross is planar; we say that the graph can be drawn in the plane.

  23. Planarity:

  24. Planarity: B A D C

  25. Planarity: B A D C

  26. Planarity:

  27. Planarity:

  28. Planarity:

  29. Planarity: Euler's formula • The problem of determining which graphs are planar was indirectly addressed by Euler in the formula V-E+F=2, where V is the number of vertices, E edges and F faces.

  30. Isomorphisms: • Two graphs that have the same relationships but possibly different drawings are called isomorphic. • http://www.planarity.net/#

  31. Counting: • One family of graphs is called the complete graphs, or cliques. These graphs include all possible edges between each pair of vertices. • An interesting exercise that illustrates some of the fun of graph theory is to count the number of edges in a clique on n vertices.

  32. Counting:

  33. Counting:

  34. Counting:

  35. Counting: • The number of edges in the (n+1)-clique is the number of edges in the (n)-clique + n.

  36. Counting: • There are n vertices. Each vertex is adjacent to all the others. There are (n-1) other vertices, so that makes n(n-1) edge-ends. Each edge has two ends, so there are n(n-1)/2 edges in an n-clique.

  37. Coding:

  38. Coding:

  39. Coding:

  40. Coding:

  41. Coding:

  42. Coding:

  43. Coding:

  44. Coding:

  45. Coding:

  46. Coding:

  47. Coding: 0 1

  48. Coding: 00 01 10 11

  49. Coding: 000 001 101 010 110 111

  50. Coding: 0000 0001 1001 1101 0010 0110 1111 1110

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