1. Graph Theory: Traveling Salesman Problem (TSP) E 3 Teacher Summer Research Program 2005 Texas A & M University June 29, 2005

2. TAKS Objectives 9 th & 10 th Grade Objective 1 – Ab1A, B, C, D & E Objective 2 – Ab2A & D, Ab3A & B Objective 3 – Ac1A & C Objective 6 – 8.7D Objective 10 - 8.14A, B & C, 8.15A, 8.16A & B

3. Material GEO Board Yarn Rubber bands Colored pencils Grid paper Graphing calculator

4. Vocabulary Nodes (vertices) Edges (arcs) Degree Adjacent Path

5. Vocabulary Length Circuit Simple Graph Complete Graph

6. Nodes (vertices)

7. Land Phone numbers People Junction points (electric circuits) Atom Chess players Companies or industries

8. Edges (arcs)

9. Bridges connecting land Calls made from one number to another connecting phone numbers Relationships or acquaintances connecting by people Wires connecting junction points (electric circuits) Bonds between atoms connecting atoms Matches connecting chess players (chess tournament) Transactions connecting companies or industries

10. Degree A F H G B E D C

11. Node A is of two degrees Node B is of two degrees Node C is of six degrees Node D is of three degrees How many degrees are nodes E, F, G and H?

12. Adjacent A F H G B E D C

13. Node A is adjacent to Nodes C and H Node B is adjacent to Nodes F and H Node C is adjacent to Nodes A, D, E, F, G and H Node D is adjacent to Nodes C, E and H What nodes are adjacent to nodes E, F, G and H?

14. Path A B C D E F

15. A B C D E F

16. A B C D E F

17. No Path A B C D E F

18. Draw two different paths. A B C D E F

19. Length A B C D E F

20. A B C D E F

21. A B C D E F

22. A B C D E F

23. Circuit Point A is the starting point C B D AE

24. C B D AE

25. C B D AE

26. C B D A E

27. Simple Graphs

28. Not, Simple Graphs

29. Complete Graphs

30. Draw a complete graph with six nodes. Draw a complete graph with seven nodes

31. Petroleum Delivers 745 325 490 825 520 565 370 570 565 380 S T N M C T Texas S South Carolina N New York M Minnesota C Colorado

32. 1 Node

33. 2 Nodes

35. 3 Nodes

42. 2 Routes

43. 4 Nodes

53. 6 Routes

54. 2 Node Node(s) Process Column Route(s) 21

55. 3 Nodes Nodes Process Column Route(s) 32 21

56. 4 Nodes Nodes Process Column Route(s) 43 32 21

57. 2 Nodes Node(s) Process Column Route(s) 22-11

58. 3 Nodes Nodes Process Column Route(s) 33-12 21

59. 3 Nodes Nodes Process Column Route(s) 33-12 22-11

60. 4 Nodes Nodes Process Column Route(s) 44-13 32 21 nr1r1

61. 4 Nodes Nodes Process Column Route(s) 44-13 33-12 21 nr1r1

62. 4 Nodes Nodes Process Column Route(s) 44-13 33-12 22-11 nr1r1

63. 4 Nodes Nodes Process Column Route(s) 44-13 33-12 22-11 nn-1r1r1

64. Function Rule r 1 equals number of routes from each remaining nodes r 1 = n - 1

65. 4 Nodes 3 2 1

66. Total Original Routes 4 Nodes 3 Routes 1 2 3 2 Routes 2 1 2 1 2 1 1 Route 1 1 1 1 1 1

67. Total Original Routes 3 x 2 x 1 = 6 or (4-1)!=6

68. Function Rule r 2 equals number of unique routes

69. Petroleum Delivers 745 325 490 825 520 565 370 570 565 380 S T N M C T Texas S South Carolina N New York M Minnesota C Colorado