Transcript Document

By: Drew Moen
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
Applications
Programming
Engineering
Communications
Circuitry
Social Networks
Shortest Path
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 nxn or nxm
checkerboard to visit every vertex once
Setup
Create a graph
 Model graph with a matrix

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
Graph
Matrix
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]
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]
Matrix Application
A2=All locations a knight can travel in two
moves
A3= three moves, A4, A5, A6…
C2= [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]
More Moves
C3=
[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]
C4= [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]
C5= [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]
Patterns
[0
C11=
C10 =
[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
[496
[0
[496
[0
[528
[0
[528
[0
0
256
0
240
0
272
0
256
0
496
0
496
0
0
0
528
0
528
256
0
240
0
0
0
272
0
256
0
496
0
528
0
496
0
528
0
0
240
0
256
0
256
0
272
0
496
0
528
0
0
0
496
0
528
240]
0 ]
256]
0 ]
0 ]
0 ]
256]
0 ]
272]
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 ]
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
<http://aix1.uottawa.ca/~jkhoury/graph.htm>.
Ramas, Amy. "Art of Knight Graph." knight_tour. 04 July 2007. 16
Dec 2008 <http://wiki.phiepsilon.org/doku.php?id=knight_tour>.
"Graph Theory & Knight's Tour." 18 Dec 2008
<http://en.wikipedia.org>.
Farmer, Jesse. "Graph Theory." 31 July 2007. 15 Dec 2008
<http://20bits.com/articles/graph-theory>.
Hickethier, Don. Q&A interview. 17 Dec 2008.