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Math Notes

# Math Notes - Introduction to Mathematics Jordan Davidson...

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Introduction to Mathematics- Jordan Davidson Professor Swets- MCS 212 No Cumulative Final exam HOSTS Program incentive- If you successfully complete it, Dr. Swets will add one letter grade to the test of your choice. HOSTS Phone #’s- (325) 481-3353 -3355 Course Topics-5 1. – Paths and Networks 2. – Voting Methods 3. – Game Theory 4. – Apportionment 5. – Probability Thursday January 18, 2007- Section 6.1, Problems 8, 12, 16, 20, and 26. Graph Theory- Graph has only dots and lines. Consists of vertices and edges. Edges may only meet at vertices. Terminology:- Even Vertices – attached to an even number of edges is even . Odd Vertices- A vertex attached to an odd number of edges is odd . An edge that starts and ends at the same vertex is a loop . A path is a collection of edges, in order, that starts at some vertex and ends at a (possibly the same) vertex. We usually mark the edges in some way to indicate the order they are traversed. If the starting and ending vertices are the same, the path is a circuit. If for any two vertices in a graph there exists a path starting at one of the vertices and ending at the other, the graph is connected . If there exist two points in a graph for which there is not path starting at one of the vertices and ending at the other, the graph is disconnected . A component of a graph is a connected part of a graph. A Bridge is an essential edge that connects the graph. If taken away, the graph becomes disconnected. So a bridge is an edge which, if it were removed, would increase the number of components in a graph. A path that includes each edge in a graph exactly once is an Euler or Eulerian Path . A circuit that includes each edge in a graph exactly once is an Euler or Eulerian Circuit . Euler’s Theorem - A graph has an Euler circuit if an only if 1. It must be connected; and 2. every vertex is even.

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A graph has an Euler path if and only if 1. It is connected; and 2. it either has no odd vertices or exactly two odd vertices, and Euler path must start at one of them and end at the other. Fleury’s Algorithm 1. Make sure an Euler circuit exists (graph connected and all vertices even). 2. Start at any vertex. 3. Travel through and mark an edge if either a. It is not a bridge for the untravelled part of the graph; or b. There is no other alternative 4. Continue until done. Thursday, January 18, 2007 - Dr. Swets – Eulerization - Identify all the odd vertices. (there will be an even number of them.) Match the odd vertices in nearby pairs. Duplicate the edges along the shortest path between each pair of vertices you matched up. Note: You may only duplicate existing edges- never create a new edge. Applications of Euler Circuits The uses of them include Garbage collection routes Mail Delivery Bus routes Snow plows Street sweeping The Traveling Salesman Problem Here is a graph with five cities (Vertices) and you want to find the cheapest way to visit all five cities Hamilton Circuit - A circuit that begins at a vertex, passes through every other vertex
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Math Notes - Introduction to Mathematics Jordan Davidson...

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