Graph Theory By Amy C. and John M.

What is Graph Theory? Graph Theory is the study of graphs It concerns the relationship among lines and points. It is also one of the most important areas of applied mathematics. History

Why is Graph Theory Important??? Graph theory is important because of its’ applications. Can be used to map out cellular telephone networks by using each persons phone and antenna as nodes and the segments would be the calls to one another. It is used to model problems in virtually all the applied sciences.

Basic Definitions Graph: A graph is collection of nodes in which various pairs of nodes are connected by line segments. Vertex: This typically means a corner or a point where lines meet. Edge: This is a line segment that connects two vertices.

Basic Definitions (cont.) Directed Graph: Line segments have arrowheads on them indicating direction. Undirected Graph: Line segments have no arrowheads on them. Weighted Graph: These are digraphs in which numbers called weights are attached to the directed edges. Path: A path is a sequence of vertices in which each vertex is adjacent to the next one.

More Definitions Length: The number of edges in a path is called the length. Cycle: This is a path of length greater than one that begins and ends at the same vertex. Loop: An edge with both ends the same. Adjacency Matrix: This is an nxn matrix where n is the number of vertices in the graph.

Matrix Example The matrix A is called the adjacency matrix of the graph. The adjacency matrix for the graph is given by

Examples in the Future The programming example we will be doing in a later presentation is called Graph Searching. To search a graph, you need to visit all the vertices in a systematic order. Illustrative examples that we will be presenting are called “Instant Insanity”, the “Shortest Path”, and the “Four Color Theorem Problem”.