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Insight into Network Structures: Exploring Graph Theory Applications with Practical Examples

Dive into the world of graph theory, a field within mathematics and computer science that analyzes graphs to model relationships between objects. Discover the historical significance of graph theory, such as Leonhard Euler's Seven Bridges of Königsberg problem. Explore how graph theory applies to various domains like transportation networks, chemistry, and social sciences. Learn about functional forms and their role in providing compact mathematical descriptions of phenomena. Join us in unraveling the complexity of networks and their properties with real-world examples.

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Insight into Network Structures: Exploring Graph Theory Applications with Practical Examples

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  1. Introduction to Networks HON207

  2. Graph Theory • In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. "Graphs" in this context are not to be confused with "graphs of functions" and other kinds of graphs

  3. Graph Theory • One of the first results in graph theory appeared in Leonhard Euler's paper on Seven Bridges of Königsberg, published in 1736. It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements.

  4. The city of Königsberg was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point. In 1736, Leonhard Euler proved that it was not possible.Circa 1750, the prosperous and educated townspeople allegedly walked about on Sundays trying to solve the problem. → →

  5. Euler’s solution • Euler realized that the problem could be solved in terms of the degrees of the nodes. The degree of a node is the number of edges touching it; in the Königsberg bridge graph, three nodes have degree 3 and one has degree 5. Euler proved that a circuit of the desired form is possible if and only if there are no nodes of odd degree

  6. Practice your Latin

  7. Contemporary Graph theory Applied Graph Theory is related to finding a measurable quantity within the network, for example, for a transportation network, the level of vehicular flow within any portion of it. Graph theory is also used to study molecules in chemistry and physics, and social networks in social sciences.

  8. What are we going to do? • We will explore some networks and their properties, in particular, their functional forms.

  9. Functional Form • A functional form is a mathematical statement of the relationship between variables in a model

  10. Why? • Developing compact mathematical descriptions of phenomena is an important step in the development of theoretical explanations. • For example: • Tyco --> Keppler --> Newton

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