7: Graph Theory

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7.1: Basic Graphs and Graphs Structure
This section introduces graph theory, defining graphs, vertices, and edges, and distinguishing simple graphs from multigraphs. It explores vertex classification, degrees, and various graph types like complete and isomorphic graphs. Key concepts include walks, trails, paths, and graph connectivity, with applications in real-world scenarios such as Page Rank and fraud detection.
7.2: Euler Circuits and Eulerization of Graph
This section covers Euler paths and circuits, key concepts in graph theory from the Konigsberg Bridge Problem. An Euler path visits every edge once with distinct starting and ending vertices, while an Euler circuit starts and ends at the same vertex. A graph can have these if it meets specific vertex degree conditions.
7.3: Hamiltonian Circuits and the Traveling Salesman Problem
This section explores Hamilton paths and circuits, their significance in graph theory, and their application in optimizing routes like school buses in Boston, saving $5 million annually. It covers methods to identify these paths and circuits, including the brute force method and the Nearest Neighbor algorithm, which provides approximate solutions for problems like the Traveling Salesman Problem.
7.4: Trees
This section covers trees in graph theory, defining them as connected acyclic graphs and exploring their significance in applications like family trees and computer networks. It discusses spanning trees, including methods for their construction, emphasizing Kruskal's algorithm for finding minimum spanning trees to minimize connection costs. Key characteristics of spanning trees are highlighted, including their cycle-free nature and connection of all vertices.

Thumbnail: A pictorial representation of a simple linear program with two variables and six inequalities. The set of feasible solutions is depicted in yellow and forms a polygon, a 2-dimensional polytope. The linear cost function is represented by the red line and the arrow: The red line is a level set of the cost function, and the arrow indicates the direction in which we are optimizing. (CC0; Ylloh via Wikipedia)

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