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Published byKelly Nash Modified over 9 years ago
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5.4 Graph Models (part I – simple graphs)
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Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life problems is called modeling
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Example: Using graph to represent the picture of the seven bridges of Konigsberg Vertices represent the banks and lands Edges represent the bridges
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Draw a graph to model this city More examples: (textbook, page 170: Bridges of Madison County) (www.coursecompass.com, #23, 25)
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5.5 Euler’s Theorems
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Who is Leonhard Euler?
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Euler’s Theorem 1 a) If a graph has any odd vertices, then it cannot have an Euler Circuit b) If a graph is connected and every vertex is an even vertex, then it has at least one Euler circuit
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Euler’s Theorem 2 a) If a graph has more than two odd vertices then it cannot have an Euler path b) If a graph is connected and has exactly two odd vertices, then it has an Euler path, starting at one odd vertex and ending at the other.
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Determine if the graph has an Euler circuit, an Euler path or neither of these No, neither Yes, an Euler path Yes, an Euler circuit
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Euler’s Theorem 3 a) The sum of the degrees of all the vertices of a graph equals twice the number of edges. b) A graph always has an even number of odd vertices
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5.4 Graph Models (part II –graphs)
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Model for a security guard
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Model for the mail carrier
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Look at page 177: Models for security guard and mail carrier
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5.6 Fleury’s Algorithm
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Algorithm on finding an Euler’s path or circuit Use a vertex to start (make sure you choose the odd vertex if the graph has an Euler path) Do not go through any bridge of the un- traveled part of the graph unless it is the only way you can go
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