1. SOME APPLICATIONS OF GRAPH THEORY
Bijo S Anand
Assistant Professor
Department of Mathematics
S.N. College, Punalur
17-02-2017
Bijo S AnandAssistant ProfessorDepartment of MathematicsS.N. College, PunalurSOME APPLICATIONS OF GRAPH THEORY 17-02-2017 1 / 16
2. Königsberg Bridge Problem
Königsberg Bridge Problem
Figure 1
Problem: The problem was to start at any of the four land areas A, B,
C, or D, walk over each of the seven bridges exactly once, and return
to the starting point (without swimming across the river).
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3. Königsberg Bridge Problem
Euler represented this situation by means of a graph. The vertices
represent the land areas and the edges represent the bridges.
Figure 2
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4. Königsberg Bridge Problem
Euler represented this situation by means of a graph. The vertices
represent the land areas and the edges represent the bridges.
Figure 2
The Konigsberg bridge problem is the same as the problem of drawing
gures without lifting the pen from the paper and without retracing a
line. He proved that such a drawing exist only when all the vertices are
of even degree.
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5. A graph is a pair of sets (V , E), where V is the set of vertices and E
is a set of edges.
Figure 3
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6. Figure 4
Figure 5
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7. Utilities Problem
Utilities Problem
Figure 6
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8. Utilities Problem
We can represent this as a graph in the following way.
Figure 7
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9. Electrical Network Problems
Electrical Network Problems
Figure 8
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10. Four Colour Theorem
Four Colour Theorem
We want to colour a map of countries so that no two adjacent
countries get the same colour.
Figure 9
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11. Four Colour Theorem
The four-color conjecture states that four colors are sucient for
coloring any atlas (a map on a plane) such that the countries with
common boundaries have dierent colors.
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12. Digital Graphs
Digital Graphs
Figure 10
Every page in the internet could be a vertex in a graph, and whenever
there is a link between two pages, there is an edge between the
corresponding vertices.
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13. Digital Graphs
Early web search engines had a very big problem: they could search
the web for a particular keyword, but they couldn't determine whether
a page is good or just spam.
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14. Digital Graphs
Early web search engines had a very big problem: they could search
the web for a particular keyword, but they couldn't determine whether
a page is good or just spam.
Google found a solution to this: any page that is very good will have
many other pages linking to it. Pages that are rarely visited, or not
very interesting, will be very lonely in the internet graph with only
few other pages linking to it. This gives a way to rank websites and
allows Google to display the best results at the beginning.
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15. Digital Graphs
Figure 11
There is a another digital graph, of which you yourself are a part:
Facebook.
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16. Digital Graphs
Figure 11
There is a another digital graph, of which you yourself are a part:
Facebook.
All the users form vertices and whenever two users are friends they are
linked by an edge. Graph theory can help web developers improve the
performance of social networking sites, and it can help us understand
Facebook better.
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17. Digital Graphs
Figure 12
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18. Digital Graphs
References I
[1] Bondy, John Adrian and Murty, Uppaluri Siva Ramachandra, Graph
Theory with Applications, Citeseer, 290, 1976
[2] Harary, Frank, Graph theory, Addison-Wesley, Reading, MA. 1969
[3] Narsingh Deo, Graph Theory with Applications to Engineering and
Computer Science, Prentice-Hall Inc.2000
[4] Vasudev C, Graph theory with applications, New Age International,
2006
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19. Digital Graphs
THANK YOU
THANK YOU
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