![Page 1: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/1.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Konigsberg, Euler and the origins of graph theory
Philip Puylaert
February 2014
![Page 2: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/2.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Konigsberg, East Prussia
capital of East Prussia (1457–1945)
Pregel river
university
birth place of Immanuel Kant, David Hilbert, Kathe Kollwitz
destroyed at the end of World War II
![Page 3: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/3.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
![Page 4: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/4.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
![Page 5: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/5.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
![Page 6: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/6.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges of Konigsberg
![Page 7: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/7.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges of Konigsberg
![Page 8: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/8.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Leonhard Euler
Basel 1707 – St.-Petersburg 1783
professor at 20
enormously productive
influence found everywhere in mathand physics
most famous formula: 1 + e iπ = 0
![Page 9: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/9.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges problem
A
B
C
D
Definitions
graph
vertices (singular: vertex) — edges
order of a vertex
![Page 10: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/10.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges problem
A
B
C
D
Definitions
graph
vertices (singular: vertex) — edges
order of a vertex
![Page 11: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/11.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges problem
A
B
C
D
Definitions
graph
vertices (singular: vertex) — edges
order of a vertex
![Page 12: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/12.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
When can you take the desired walk?
A
1
2
3
4
vertex of even order
A
1
23
vertex of odd 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
![Page 13: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/13.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
When can you take the desired walk?
A
1
2
3
4
vertex of even order
A
1
23
vertex of odd 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
![Page 14: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/14.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
When can you take the desired walk?
A
1
2
3
4
vertex of even order
A
1
23
vertex of odd 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
![Page 15: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/15.jpg)
History of Konigsberg The 7 bridges of Konigsberg 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
BC
1
2
3
A B
CD
1
2
3
4 5
![Page 16: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/16.jpg)
History of Konigsberg The 7 bridges of Konigsberg 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
BC
1
2
3
A B
CD
1
2
3
4 5
![Page 17: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/17.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Back to the 7 bridges problem
A
B
C
D
the order of A is 3
the order of B is 4
the order of C is 3
the order of D is 3
Conclusion
The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.
![Page 18: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/18.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Back to the 7 bridges problem
A
B
C
D
the order of A is 3
the order of B is 4
the order of C is 3
the order of D is 3
Conclusion
The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.
![Page 19: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/19.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Back to the 7 bridges problem
A
B
C
D
the order of A is 3
the order of B is 4
the order of C is 3
the order of D is 3
Conclusion
The graph of the 7 bridges problem is not traversable.It’s impossible to take a walk crossing every bridge exactly once.
![Page 20: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/20.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 1: traffic
![Page 21: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/21.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 2: social networks
![Page 22: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/22.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 2: social networks
![Page 23: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/23.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 3: ranking of search results by Google
each vertex represents a web pagearrow D → A means: page D contains a link to page A
![Page 24: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/24.jpg)
History of Konigsberg The 7 bridges of Konigsberg 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 avertex
you and Euler solved the 7 bridges problem by proving when agraph is traversablethe Konigsberg graph is not traversable
some applications of graph theory, e.g. traffic, social networks
![Page 25: Königsberg, Euler and the origins of graph theory](https://reader034.vdocuments.us/reader034/viewer/2022052307/555095e2b4c9058b208b45da/html5/thumbnails/25.jpg)
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
More information?
Reinhard Diestel, Graph Theory (3rd edition), Springer Verlag,2005www.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 Bridgewww.youtube.com/watch?v=BK kYjFWWX0