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5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life.

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Presentation on theme: "5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life."— Presentation transcript:

1 5.4 Graph Models (part I – simple graphs)

2 Graph is the tool for describing real-life situation. The process of using mathematical concept to solve real-life problems is called modeling

3 Example: Using graph to represent the picture of the seven bridges of Konigsberg Vertices represent the banks and lands Edges represent the bridges

4 Draw a graph to model this city More examples: (textbook, page 170: Bridges of Madison County) (www.coursecompass.com, #23, 25)

5 5.5 Euler’s Theorems

6 Who is Leonhard Euler?

7 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

8 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.

9 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

10 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

11 5.4 Graph Models (part II –graphs)

12 Model for a security guard

13 Model for the mail carrier

14 Look at page 177: Models for security guard and mail carrier

15 5.6 Fleury’s Algorithm

16 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

17 12345678 AB D F E C

18 123 10 456789 1514131211 2721201918 1617 26 25242322


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