königsberg, euler and the origins of graph theory
DESCRIPTION
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.TRANSCRIPT
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
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
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Nowadays: Kaliningrad
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges of Konigsberg
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
The 7 bridges of Konigsberg
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
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
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
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
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
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
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
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
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
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.
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.
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.
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 1: traffic
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 2: social networks
History of Konigsberg The 7 bridges of Konigsberg Applications of graph theory Summary & further reading
Application 2: social networks
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
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
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