1. By: Drew Moen

2. Graph Theory History Leonhard Euler - founder The Seven Bridges of Königsberg Cross every Bridge once Change the city into a graph Change the graph into a matrix

3. Applications Programming Engineering Communications Circuitry Social Networks Shortest Path

4. Knight’s Tour Hamilton Path A path that visits every vertex on a graph one time Knight’s Tour A path that a knight takes on a n x n or n x m checkerboard to visit every vertex once Setup Create a graph Model graph with a matrix

5. Purpose Finding new ways to solve for a knight’s tour Figuring out where a knight can arrive with a restricted amount of moves Finding out how many moves a knight needs to get anywhere on the board

6. Graph

7. Matrix Three by Three C=[0 0 0 0 0 1 0 1 0] [0 0 0 0 0 0 1 0 1] [0 0 0 1 0 0 0 1 0] [0 0 1 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0] [1 0 0 0 0 0 1 0 0] [0 1 0 0 0 1 0 0 0] [1 0 1 0 0 0 0 0 0] [0 1 0 1 0 0 0 0 0] Four by Four B=[0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0] [1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1] [0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0] [0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0] [1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0] [0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0]

8. Matrix Application A 2 =All locations a knight can travel in two moves A 3 = three moves, A 4, A 5, A 6 … C 2 = [2 0 1 0 0 0 1 0 0] [0 2 0 1 0 1 0 0 0] [1 0 2 0 0 0 0 0 1] [0 1 0 2 0 0 0 1 0] [0 0 0 0 0 0 0 0 0] [0 1 0 0 0 2 0 1 0] [1 0 0 0 0 0 2 0 1] [0 0 0 1 0 1 0 2 0] [0 0 1 0 0 0 1 0 2]

9. More Moves C 3 = [0 1 0 1 0 3 0 3 0] [1 0 1 0 0 0 3 0 3] [0 1 0 3 0 1 0 3 0] [1 0 3 0 0 0 1 0 3] [0 0 0 0 0 0 0 0 0] [3 0 1 0 0 0 3 0 1] [0 3 0 1 0 3 0 1 0] [3 0 3 0 0 0 1 0 1] [0 3 0 3 0 1 0 1 0] C 4 = [6 0 4 0 0 0 4 0 2] [0 6 0 4 0 4 0 2 0] [4 0 6 0 0 0 2 0 4] [0 4 0 6 0 2 0 4 0] [0 0 0 0 0 0 0 0 0] [0 4 0 2 0 6 0 4 0] [4 0 2 0 0 0 6 0 4] [0 2 0 4 0 4 0 6 0] [2 0 4 0 0 0 4 0 6] C 5 = [0 6 0 6 0 10 0 10 0 ] [6 0 6 0 0 0 10 0 10] [0 6 0 10 0 6 0 10 0 ] [6 0 10 0 0 0 6 0 10] [0 0 0 0 0 0 0 0 0 ] [10 0 6 0 0 0 10 0 6 ] [ 0 10 0 6 0 10 0 6 0 ] [10 0 10 0 0 0 6 0 6 ] [ 0 10 0 10 0 6 0 6 0 ]

10. Patterns [272 0 256 0 0 0 256 0 240] [0 272 0 256 0 256 0 240 0 ] [256 0 272 0 0 0 240 0 256] [0 256 0 272 0 240 0 256 0 ] [0 0 0 0 0 0 0 0 0 ] [0 256 0 240 0 272 0 256 0 ] [256 0 240 0 0 0 272 0 256] [0 240 0 256 0 256 0 272 0 ] [240 0 256 0 0 0 256 0 272] [ 0 496 0 496 0 528 0 528 0 ] [496 0 496 0 0 0 528 0 528] [0 496 0 528 0 496 0 528 0 ] [496 0 528 0 0 0 496 0 528] [0 0 0 0 0 0 0 0 0 ] [528 0 496 0 0 0 528 0 496] [0 528 0 496 0 528 0 496 0 ] [528 0 528 0 0 0 496 0 496] [0 528 0 528 0 496 0 496 0 ] C 11 = C 10 =

11. Work’s Cited Rosen, Kenneth H.. Discrete Mathematics and Its Applications. Fifth. New York, NY: McGraw-Hill, 2003. Strang, Gilbert. Introduction to Linear Algebra. Third. Wellesley MA: Wellesley-Cambridge Press, 2003. Houry, J K.. "Application to Graph theory." 11 Nov 2008. Ramas, Amy. "Art of Knight Graph." knight_tour. 04 July 2007. 16 Dec 2008. "Graph Theory & Knight's Tour." 18 Dec 2008. Farmer, Jesse. "Graph Theory." 31 July 2007. 15 Dec 2008. Hickethier, Don. Q&A interview. 17 Dec 2008.