Graph theory

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Graph Theory is about analyzing Graphs. A graph is a group of points and lines connecting each other, to make a picture of such a route. It is less detailed than a map and is used to find answers.

For example:

• What is the best way for a mailman to get to all of the houses in the area in the least amount of time? The points could represent street corners and lines could represent the houses along the street. (see Chinese postman problem)

• A salesman has to visit different customers, but wantas to keep the distance travelled as small as possible. Find a way so they can do it. This problem is known as Travelling Salesman Problem (and often abbreviated TSP). It is among the hardest problems to solve. An exact solution requires to try all possible routes, to find which is shortest.

• How many colors do you need to color a map? The points could represent the different areas and the lines could represent that two areas are neighboring. (look at the Four color theorem)

• Can you sketch a drawing in one closed line? The lines of your drawing are the lines of the graph and when two or more lines collide, there is a point in the graph. The task is now to find a way through the graph using each line one time. (look at Seven Bridges of Königsberg)

[edit] Different kinds of Graphs

• Graph theory has many aspects. Graphs can be directed on undirected. An example of a directed graph would be the system of roads in a city. Some streets in the city are one way streets. This means, that on those parts there is only one direction to follow.
• Graphs can be weighted. An example would be a road network, with distances, or with tolls (for roads).
• The nodes (the circles in the schematic) of a graph are called vertices. The lines connecting the nodes are called edges. There can be no line between two nodes, there can be one line, or there can be multiple lines.

[edit] Graph theory in perspecive

Graph theory is an important part of mathematics and computer science. To many such problems, exact solutions do exist. Many times however, they are very hard to calculate. Therefore, very often, approximations are used. There are two kinds of such approximations, Monte-Carlo algorithms and Las-Vegas algorithms.

Graphs are normally represented by two different sets, typically set a graph G would be represented as the collection of the sets V and E. The set V is a discrete set containing all verticies of the graph. The set E is a binary set, whose pairwise elements are elements of set V. Each pair in set E represents an edge connecting two verticies.

If for all elements v1 and v2 of the set V, if {v1, v2} and {v2, v1} are both elements of E then the graph G is considered a Complete Graph.

I would move forth to define a Path, a Walk, A weighted graph, a directional graph. Then make a comment about how all trees are just subsets of graphs. You could also prove that all connected graphs can be represented as a Tree. There is a lot of graph theory not explained here.

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