A slidecast explaining the origins of graph theory and the solution to the 7 bridges problem of Königsberg. I discuss some modern applications of graph theory too.

1. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
K¨nigsberg, Euler and the origins of graph theory
o
Philip Puylaert
February 2014

2. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
K¨nigsberg, East Prussia
o
capital of East Prussia (1457–1945)
Pregel river
university
birth place of Immanuel Kant, David Hilbert, K¨the Kollwitz
a
destroyed at the end of World War II

3. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Nowadays: Kaliningrad
Applications of graph theory
Summary & further reading

4. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Nowadays: Kaliningrad
Applications of graph theory
Summary & further reading

5. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Nowadays: Kaliningrad
Applications of graph theory
Summary & further reading

6. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading

7. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading

8. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Leonhard Euler
Basel 1707 – St.-Petersburg 1783
professor at 20
enormously productive
influence found everywhere in math
and physics
most famous formula: 1 + e iπ = 0
Summary & further reading

9. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
The 7 bridges problem
Applications of graph theory
Summary & further reading

10. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
The 7 bridges problem
A
B
C
D

11. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
The 7 bridges problem
A
B
C
Definitions
graph
vertices (singular: vertex) — edges
order of a vertex
D

12. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
When can you take the desired walk?
1
4
A
2
3
vertex of even order
Summary & further reading

13. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
When can you take the desired walk?
1
4
A
1
2
3
A
3
2
vertex of odd order
vertex of even order

14. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
When can you take the desired walk?
1
4
A
1
2
A
3
3
2
vertex of odd order
vertex of even order
The graph is traversable
if all vertices have even order
→ Euler tour, a closed walk
if exactly 2 vertices have odd order
→ use them to start and finish your walk

15. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Examples of traversable graphs
The graph is traversable
if all vertices have even order
→ Euler tour, a closed walk
if exactly 2 vertices have odd order
→ use them to start and finish your walk
A
1
3
C
2
B
Summary & further reading

16. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
Examples of traversable graphs
The graph is traversable
if all vertices have even order
→ Euler tour, a closed walk
if exactly 2 vertices have odd order
→ use them to start and finish your walk
A
1
3
C
2
1
B
4
A
5
2
D
3
C
B

17. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Back to the 7 bridges problem
A
B
C
D
Summary & further reading

18. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
Back to the 7 bridges problem
the order of A is 3
A
the order of B is 4
the order of C is 3
the order of D is 3
B
C
D

19. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
Back to the 7 bridges problem
the order of A is 3
A
the order of B is 4
the order of C is 3
the order of D is 3
B
D
C
Conclusion
The graph of the 7 bridges problem is not traversable.
It’s impossible to take a walk crossing every bridge exactly once.

20. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Application 1: traffic
Applications of graph theory
Summary & further reading

21. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Application 2: social networks
Applications of graph theory
Summary & further reading

22. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Application 2: social networks
Applications of graph theory
Summary & further reading

23. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
Application 3: ranking of search results by Google
each vertex represents a web page
arrow D → A means: page D contains a link to page A

24. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
Summary
What have you learned in this slidecast?
basic concepts of graph theory: graph, vertex, edge, order of a
vertex
you and Euler solved the 7 bridges problem by proving when a
graph is traversable
the K¨nigsberg graph is not traversable
o
some applications of graph theory, e.g. traffic, social networks

25. History of K¨nigsberg
o
The 7 bridges of K¨nigsberg
o
Applications of graph theory
Summary & further reading
More information?
Reinhard Diestel, Graph Theory (3rd edition), Springer Verlag,
2005
www.math.ubc.ca/~solymosi/2007/443/GraphTheoryIII.pdf
Fred Buckley, A Friendly Introduction to Graph Theory,
Prentice Hall, 2002
Glen Gray, Graph Theory 1 — Intro via Konigsberg Bridge
www.youtube.com/watch?v=BK kYjFWWX0