# 6: Graph Theory


Graph theory deals with routing and network problems and if it is possible to find a “best” route, whether that means the least expensive, least amount of time or the least distance. Some examples of routing problems are routes covered by postal workers, UPS drivers, police officers, garbage disposal personnel, water meter readers, census takers, tour buses, etc. Some examples of network problems are telephone networks, railway systems, canals, roads, pipelines, and computer chips.

• 6.1: Graph Theory
Some definitions are important to understand before delving into Graph Theory: (1) A graph is a picture of dots called vertices and lines called edges. (2) An edge that starts and ends at the same vertex is called a loop. (3) If there are two or more edges directly connecting the same two vertices, then these edges are called multiple edges. (4) If there is a way to get from one vertex of a graph to all the other vertices of the graph, then the graph is connected, otherwise it is disconnected.
• 6.2: Networks
A network is a connection of vertices through edges. The internet is an example of a network with computers as the vertices and the connections between these computers as edges.
• 6.3: Euler Circuits
Leonhard Euler first discussed and used Euler paths and circuits in 1736. Rather than finding a minimum spanning tree that visits every vertex of a graph, an Euler path or circuit can be used to find a way to visit every edge of a graph once and only once. This would be useful for checking parking meters along the streets of a city, patrolling the streets of a city, or delivering mail.
• 6.4: Hamiltonian Circuits
The Traveling Salesman Problem (TSP) is any problem where you must visit every vertex of a weighted graph once and only once, and then end up back at the starting vertex. Examples of TSP situations are package deliveries, fabricating circuit boards, scheduling jobs on a machine and running errands around town.
• 6.5: Exercises

Thumbnail: Königsberg graph (CC BY-SA 3.0; Booyabazooka via Wikipedia)

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