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

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Presentation transcript:

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

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 A, B & C, 8.15A, 8.16A & B

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

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

Vocabulary Length Circuit Simple Graph Complete Graph

Nodes (vertices)

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

Edges (arcs)

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

Degree A F H G B E D C

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?

Adjacent A F H G B E D C

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?

Path A B C D E F

A B C D E F

A B C D E F

No Path A B C D E F

Draw two different paths. A B C D E F

Length A B C D E F

A B C D E F

A B C D E F

A B C D E F

Circuit Point A is the starting point C B D AE

C B D AE

C B D AE

C B D A E

Simple Graphs

Not, Simple Graphs

Complete Graphs

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

Petroleum Delivers S T N M C T Texas S South Carolina N New York M Minnesota C Colorado

1 Node

2 Nodes

3 Nodes

2 Routes

4 Nodes

6 Routes

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

3 Nodes Nodes Process Column Route(s) 32 21

4 Nodes Nodes Process Column Route(s)

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

3 Nodes Nodes Process Column Route(s)

3 Nodes Nodes Process Column Route(s)

4 Nodes Nodes Process Column Route(s) nr1r1

4 Nodes Nodes Process Column Route(s) nr1r1

4 Nodes Nodes Process Column Route(s) nr1r1

4 Nodes Nodes Process Column Route(s) nn-1r1r1

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

4 Nodes 3 2 1

Total Original Routes 4 Nodes 3 Routes Routes Route

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

Function Rule r 2 equals number of unique routes

Petroleum Delivers S T N M C T Texas S South Carolina N New York M Minnesota C Colorado