Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Construct a table that displays the number of directed paths of length 1 or 2 between each pair of vertices in the graph shown.

**Understanding Directed Graph Paths** **Constructing a Table for Directed Paths** To analyze the graph and determine the number of directed paths of length 1 or 2 between each pair of vertices, we can complete a table that represents these paths. Let's break down the process with a detailed explanation. ### Graph Description The graph provided consists of 7 vertices labeled A, B, C, D, E, F, and G. Directed edges (arrows) indicate the direction from one vertex to another, outlining the possible paths. ### Table Description The table is segmented into rows and columns representing the vertices. The rows indicate the starting vertices, while the columns represent the destination vertices. The table makes it easy to identify the number of directed paths of length 1 or 2 between pairs of vertices. Each cell in the table reflects the count of such paths. ### Example Table | From \ To | A | B | C | D | E | F | G | |-----------|---|---|---|---|---|---|---| | A | 0 | 1 | 0 | 0 | 1 | 1 | 2 | | B | 0 | 0 | 0 | 0 | 0 | 1 | 1 | | C | 0 | 0 | 0 | 0 | 1 | 1 | 1 | | D | 0 | 0 | 0 | 0 | 2 | 1 | 1 | | E | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | F | 0 | 0 | 0 | 1 | 0 | 0 | 0 | | G | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ### Explanation #### Path Analysis 1. **From A to Other Vertices:** - A → B: 1 path of length 1. - A → E: 1 path of length 2 (via D). - A → F: 1 path of length 2 (via F). - A → G: 2 paths (one direct and one via C). 2. **From B to Other Vertices:** - B → F: 1 path of length 1. - B
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Transcribed Image Text:**Understanding Directed Graph Paths** **Constructing a Table for Directed Paths** To analyze the graph and determine the number of directed paths of length 1 or 2 between each pair of vertices, we can complete a table that represents these paths. Let's break down the process with a detailed explanation. ### Graph Description The graph provided consists of 7 vertices labeled A, B, C, D, E, F, and G. Directed edges (arrows) indicate the direction from one vertex to another, outlining the possible paths. ### Table Description The table is segmented into rows and columns representing the vertices. The rows indicate the starting vertices, while the columns represent the destination vertices. The table makes it easy to identify the number of directed paths of length 1 or 2 between pairs of vertices. Each cell in the table reflects the count of such paths. ### Example Table | From \ To | A | B | C | D | E | F | G | |-----------|---|---|---|---|---|---|---| | A | 0 | 1 | 0 | 0 | 1 | 1 | 2 | | B | 0 | 0 | 0 | 0 | 0 | 1 | 1 | | C | 0 | 0 | 0 | 0 | 1 | 1 | 1 | | D | 0 | 0 | 0 | 0 | 2 | 1 | 1 | | E | 0 | 0 | 0 | 0 | 0 | 0 | 0 | | F | 0 | 0 | 0 | 1 | 0 | 0 | 0 | | G | 0 | 0 | 1 | 0 | 1 | 0 | 1 | ### Explanation #### Path Analysis 1. **From A to Other Vertices:** - A → B: 1 path of length 1. - A → E: 1 path of length 2 (via D). - A → F: 1 path of length 2 (via F). - A → G: 2 paths (one direct and one via C). 2. **From B to Other Vertices:** - B → F: 1 path of length 1. - B
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