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

Three by ThreeC=[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 FourB=[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]

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

[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 ]

C11=

C10 =

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.