Graph theory

Leonard Euler (“Oiler”) 1706 - 1783

Konigsberg Bridge Problem The Konigsberg bridge problem originated in the city of Konigsberg, formerly in Germany but, now known as Kaliningrad and part of Russia, located on the river preger. The city had seven bridges, which connected two islands with the main-land via seven bridges. People staying there any way to walk over all the bridges once and only once.

Konigsberg Bridge Problem

Euler's analysis

First, Euler pointed out that the choice of route inside each land mass is irrelevant. The only important feature of a route is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms (laying the foundations of graph theory), eliminating all features except the list of land masses and the bridges connecting them. In modern terms, one replaces each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which only serves to record which pair of vertices (land masses) is connected by that bridge. The resulting mathematical structure is called a graph.

APPLICATION OF GRAPH THEORY

Everyday scenarios with an underlying application of graph theory: Using your GPS or Google Maps/Yahoo Maps, to determine a route based on user settings (quickest route/shortest route) or finding the cheapest airfare between two destinations. The destinations are vertices and their connections are edges containing information such as distance or airfare. The software finds the critical path (optimal route) based on the user settings

Graph theory is used for finding communities in networks

It is also used for ranking (ordering) hyperlinks

Google Maps

GPS to find the shortest path home ...

 Computer Network Security

 vertex coloring

Edge Coloring

Map Coloring and GSM Mobile Phone Networks