1. A Presentation By: Jillian Minuto Troy Norman Alan Leggett Group Advisor: Prof. G. Warrington Graph Theory

2. A function f is a relation such that f A x B f(x)=x 3 -7

4. …Look Familiar?

5. It’s a graph!

6. A simple graph G is a pair G=(V,E) where V is a finite set of vertices of G E is a finite set of two-element subsets of V, called the edges of G Here V = {1,2,3,4} E={ a, b, c, d } {(1,2) (1,3)(3,2)(2,4)} a b c d Route of a single bus in Burlington

7. Degree of a Vertex d(3)=2 d(4)=1 a b c d Route of a single bus in Burlington d(1)=2 d(2)=3 Since all vertices have a degree ≥ 1, this is called a connected simple graph

8. Degree of a Vertex d(1)=3 d(2)=6 d(3)=4 d(4)=1 Route of all buses in Burlington

9. Path – distinct edges and vertices Path ⇒ Trail ⇒ Walk ✔ ✗ Walk ⇏ Trail ⇏ Path Where length l is defined as the number of edges traveled Distinct edges only - Trail Walk – non-distinct edges and vertices Trail starts and ends at the same vertex - Circuit

10. Eulerian Graphs

11. Eulerian Circuit A graph that can be drawn using every edge exactly once and ends at the initial vertex

12. Eulerian Trail A graph that can be drawn using every edge exactly once and ending at a different vertex than the initial.

13. Konigsberg Bridges Asks whether or not you can cross each of the bridges once and return to the starting point.

14. Euler’s Contribution Euler was able to prove that the problem of the Konigsberg Bridges was not possible. From his findings he developed a simple theorem on whether or not a graph is Eulerian.

15. Euler’s Theorem A connected graph G has an Eulerian circuit if and only if the degree of each vertex of G is even.

16. Proof of Euler’s Theorem

17. Euler’s Theorem A connected graph G has an Eulerian circuit if and only if the degree of each vertex of G is even.

18. Homework a) You and your friend take a trip to Konigsberg, if possible design a path that allows you to see the entire city and cross each bridge once and only once. b) Suppose that one fewer bridge was built in the city of Konigsberg, as shown. Design a route that allows for you to see the whole city while crossing each bridge once and only once. c) Does it matter which bridge is removed? If not, provide an example. d) Could you add a bridge(s) that would also make this task possible? If yes, provide an example.

19. Sources: Wilson, Robin J. "Chapter 3/Eulerian Graphs." Introduction to Graph Theory. Harlow: Longman, 1996. 31-33. Print. Agnarsson, Geir, and RaymondCha Greenlaw. "Chapter 5/Eulerian Graphs." Graph Theory: Modeling, Applications, and Algorithms. Upper Saddle River, NJ: Pearson/Prentice Hall, 2007. 135-37. Print. http://www.math.lsa.umich.edu/mmss/coursesONLINE/ graph/graph1/index.html http://www.math.lsa.umich.edu/mmss/coursesONLINE/ graph/graph1/index.html http://understandingsociety.blogspot.com/2011/04/math- of-social-networks.html http://understandingsociety.blogspot.com/2011/04/math- of-social-networks.html