1. Graph Theory in Computer Science
Daniel Candeias
2012/2013

2. Contents Introduction What is graph theory? Fields of study
Applications in computer Science
Graph Operations
Common Problems

3. Introduction
With this presentation I want to show you that we can use Graph Theory in diverse fields of study.
I’ll focus in Computer Science applications, and give you a general overview about it.

4. What is Graph Theory?
Definition: The mathematical theory of the properties and applications of graphs.
Used to understand and solve many of mathematical and path problems.

5. The cells of a GSM mobile phone network
Fields of Study
Mathematical problem
The cells of a GSM mobile phone network
Other examples:
Electrical eng.
Biochemistry
Computer science
Physics
Music

6. Applications in computer Science (1)
Since computer science is not a concrete/centralized subject, we can introduce graph theory in many areas.
But where
Let’s see some examples…

7. Applications in computer Science (2)
Networks: Graph theory can be used in computer networks, for security purpose or to schematize network topologies, for example.

8. Applications in computer Science (3)
Webpage: can be represented by a direct graph. The vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B.
Facebook is based in graph theory

9. Applications in computer Science (4)
Workflow: It’s sequence of processes through which a piece of work passes from initiation to completion. Can be also represented as directed graph.

10. Applications in computer Science (5)
Neural Networks: A series of algorithms that attempt to identify underlying relationships in a set of data by using a process that mimics the way the human brain operates.

11. Applications in computer Science (6)
Google Maps:

12. Graph Operations (1)
Basic Operations

13. Graph Operations (2)
Shortest Path: is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. E.g. Dijkstra’s algorithm

14. Graph Operations (3)
Minimun Spanning tree: A tree is a connected graph without cycles with some properties:
A graph is a tree if and only if there is one and only one path joining any two of its vertices.
A connected graph is a tree if and only if every one of its edges is a bridge.
A connected graph is a tree if and only if it has N vertices and N; 1 edges.
E.g. Kruskal's Algorithm

15. Graph Operations (4)
Minimun Spanning tree

16. Graph Operations (5)
Breadth-First Search (BFS): algorithm used to search in graphs.

17. Graph Operations (6)
Depth-First Search (DFS): algorithm used to search in graphs.

18. Graph Operations (7)
DFS Vs. BFS

19. Common problems (1)
House of Santa Claus: the house in the picture can be represented as a graph, where each vertice is a node. Try to draw a the house by yourself fallowing this rules:
You have to draw a house in one line. 
You must not lift your pencil while drawing.
You must not repeat a line.

20. Common problems (2)
Three Houses, Three Utilities: there are three utilities (three vertices) and three houses (three vertices), and the purpose of this problem is to draw a line from each house to each facility without the lines ever crossing.

21. Conclusions
As we can see Graph Theory can be applied to several areas.
In Computer Science is mainly used to solve problems, or to represent scenarios, related with networks.

22. Thank you!
Daniel Candeias