5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation....
-
Upload
kelly-nash -
Category
Documents
-
view
212 -
download
0
Transcript of 5.4 Graph Models (part I – simple graphs). Graph is the tool for describing real-life situation....
![Page 1: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/1.jpg)
5.4 Graph Models(part I – simple graphs)
![Page 2: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/2.jpg)
• Graph is the tool for describing real-life situation.
• The process of using mathematical concept to solve real-life problems is called modeling
![Page 3: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/3.jpg)
Example: Using graph to represent the picture of the seven bridges of Konigsberg
Vertices represent the banks and lands
Edges represent the bridges
![Page 4: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/4.jpg)
Draw a graph to model this city
More examples: (textbook, page 170: Bridges of Madison County)(www.coursecompass.com, #23, 25)
![Page 5: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/5.jpg)
5.5 Euler’s Theorems
![Page 6: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/6.jpg)
• Who is Leonhard Euler?
![Page 7: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/7.jpg)
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
![Page 8: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/8.jpg)
• 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.
![Page 9: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/9.jpg)
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
![Page 10: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/10.jpg)
• 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
![Page 11: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/11.jpg)
5.4 Graph Models(part II –graphs)
![Page 12: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/12.jpg)
Model for a security guard
![Page 13: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/13.jpg)
Model for the mail carrier
![Page 14: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/14.jpg)
• Look at page 177: Models for security guard and mail carrier
![Page 15: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/15.jpg)
5.6Fleury’s Algorithm
![Page 16: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/16.jpg)
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
![Page 17: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/17.jpg)
1 2 3 4 5 6 7 8
A B
D
FE
C
![Page 18: 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.](https://reader035.fdocuments.in/reader035/viewer/2022070401/56649f1d5503460f94c34478/html5/thumbnails/18.jpg)
1 2 3
10
4 5 6 7 8 9
1514131211
2721201918
16 17
2625242322