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nossi ch 6

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The document discusses graphs and networks, including routing problems, routes and paths, connected graphs, weighted graphs, spanning trees, and minimal spanning trees. It provides examples and definitions of key concepts like vertices, edges, paths, circuits, connected graphs, weighted graphs, spanning trees, and minimal spanning trees. Kruskal's algorithm for finding a minimal spanning tree is also explained.

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1 of 126
A Mathematical View of Our World 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer
Chapter 6 Routes and Networks
Section 6.1 Routing Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
6.1 Initial Problem ,[object Object],[object Object]
Graphs ,[object Object],[object Object]
Graphs ,[object Object],[object Object],[object Object]
Equivalent Graphs ,[object Object]
Example 1 ,[object Object],A is connected to B, B to C, and C to D
Vertices and Edges ,[object Object],[object Object]
Example 2 ,[object Object],[object Object]
Example 2 ,[object Object],[object Object]
Example 3 ,[object Object],[object Object]
Example 3, cont’d ,[object Object],[object Object],[object Object],[object Object],[object Object]
Example 4 ,[object Object],[object Object],[object Object],[object Object]
Example 4, cont’d ,[object Object],[object Object],[object Object],[object Object]
Degree of a Vertex ,[object Object],[object Object],[object Object]
Example 5 ,[object Object]
Example 5 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example 5 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Theorem ,[object Object],[object Object],[object Object]
Example 6 ,[object Object],[object Object]
Paths ,[object Object],[object Object],[object Object],[object Object]
Example 7 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Circuits ,[object Object],[object Object]
Euler Paths ,[object Object],[object Object]
Euler Circuits ,[object Object],[object Object]
Connected Graphs ,[object Object],[object Object],[object Object]
Example 8 ,[object Object],[object Object]
Example 8, cont’d ,[object Object]
Example 9 ,[object Object],[object Object]
Example 10 ,[object Object],[object Object]
Degree Theorem ,[object Object],[object Object],[object Object],[object Object]
Euler’s Theorem ,[object Object],[object Object],[object Object]
Euler’s Theorem cont’d ,[object Object],[object Object],[object Object],[object Object],[object Object]
Euler’s Theorem cont’d ,[object Object],[object Object],[object Object],[object Object]
Example 11 ,[object Object],2 F 2 E 4 D 2 C 4 B 4 A Degree Vertex
Example 11 ,[object Object],[object Object]
Example 11, cont’d ,[object Object]
Fleury’s Algorithm ,[object Object],[object Object],[object Object],[object Object]
Fleury’s Algorithm, cont’d ,[object Object],[object Object],[object Object]
Fleury’s Algorithm, cont’d ,[object Object],[object Object]
Example 12 ,[object Object],[object Object]
6.1 Initial Problem Solution ,[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object]
Section 6.1 Assignment ,[object Object]
Section 6.2 Network Problems ,[object Object],[object Object],[object Object],[object Object],[object Object]
6.2 Initial Problem ,[object Object],[object Object],[object Object],[object Object]
Weighted Graphs ,[object Object]
Example 1 ,[object Object],[object Object],[object Object],[object Object],[object Object]
Example 1, cont’d ,[object Object],[object Object],[object Object],[object Object]
Example 1, cont’d ,[object Object]
Redundant Edges ,[object Object],[object Object],[object Object]
Subgraphs ,[object Object],[object Object]
Subgraphs ,[object Object]
Example 2 ,[object Object],[object Object],[object Object]
Example 2 ,[object Object]
Example 2 ,[object Object]
Example 2 ,[object Object]
Trees ,[object Object],[object Object],[object Object]
Trees, cont’d ,[object Object]
Example 3 ,[object Object],[object Object]
Example 3 ,[object Object]
Spanning Trees ,[object Object],[object Object],[object Object],[object Object]
Spanning Trees ,[object Object],[object Object]
Minimal Spanning Trees ,[object Object],[object Object]
Minimal Spanning Trees ,[object Object],[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object]
Kruskal’s Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Example 4 ,[object Object]
Example 4 ,[object Object],[object Object],[object Object]
Example 4 ,[object Object],[object Object],[object Object]
Example 4 ,[object Object],[object Object],[object Object]
Example 4 ,[object Object],[object Object],[object Object]
6.2 Initial Problem Solution ,[object Object]
Initial Problem Solution ,[object Object],[object Object]
Initial Problem Solution ,[object Object],[object Object]
Initial Problem Solution ,[object Object],[object Object]
Section 6.2 Assignment ,[object Object]
Section 6.3 Traveling-Salesperson Problem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
6.3 Initial Problem ,[object Object],[object Object]
Hamiltonian Circuits ,[object Object],[object Object]
Example 1 ,[object Object],[object Object]
Example 2 ,[object Object],[object Object]
Hamiltonian Circuits, cont’d ,[object Object],[object Object]
Example 3 ,[object Object]
Example 3 ,[object Object],[object Object],[object Object]
Example 3, cont’d ,[object Object],[object Object]
Example 3, cont’d ,[object Object],[object Object],[object Object],[object Object]
Complete Graphs ,[object Object]
Example 4 ,[object Object]
Example 4, cont’d ,[object Object],[object Object]
Example 4, cont’d ,[object Object],[object Object]
Example 4, cont’d ,[object Object],[object Object]
Complete Graphs ,[object Object],[object Object],[object Object],[object Object]
Example 6 ,[object Object]
Example 6, cont’d ,[object Object]
Example 6, cont’d ,[object Object],[object Object],[object Object]
Theorem on Complete Graphs ,[object Object],[object Object],[object Object]
Question: ,[object Object],[object Object],[object Object],[object Object],[object Object]
Question: ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Traveling-Salesperson Problem ,[object Object],[object Object]
Complete Weighted Graphs ,[object Object],[object Object]
Question: ,[object Object],[object Object],[object Object],[object Object],[object Object]
Question: ,[object Object],[object Object],[object Object],[object Object],[object Object]
Example 7 ,[object Object],[object Object]
Example 7, cont’d ,[object Object],[object Object]
Example 7, cont’d
Example 7, cont’d ,[object Object],[object Object],[object Object]
Example 7, cont’d ,[object Object],[object Object]
Example 7, cont’d ,[object Object]
Example 7, cont’d ,[object Object],[object Object],[object Object],[object Object]
Traveling-Salesperson, cont’d ,[object Object],[object Object],[object Object],[object Object],[object Object]
Traveling-Salesperson, cont’d ,[object Object],[object Object],[object Object],[object Object],[object Object]
6.3 Initial Problem Solution ,[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object],[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object],[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object]
Initial Problem Solution, cont’d ,[object Object],[object Object]
A fun website with Traveling Salesman Problem (TSP) Artwork: http://www.oberlin.edu/math/faculty/bosch/tspart-page.html
Section 6.3 Assignment ,[object Object]
Chapter 6 Assignments ,[object Object],[object Object],[object Object]

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nossi ch 6

  • 1. A Mathematical View of Our World 1 st ed. Parks, Musser, Trimpe, Maurer, and Maurer
  • 2. Chapter 6 Routes and Networks
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  • 124. A fun website with Traveling Salesman Problem (TSP) Artwork: http://www.oberlin.edu/math/faculty/bosch/tspart-page.html
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