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Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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See attached, answer part 2
![**Problem 6**
**Part 1.** Give the adjacency matrix for the graph G as pictured below:
![Graph with 6 vertices and 9 edges](image.png)
*Figure 2:* A graph shows 6 vertices and 9 edges. The vertices are 1, 2, 3, 4, 5, and 6, represented by circles. The edges between the vertices are represented by arrows, as follows: 4 to 3; 3 to 2; 2 to 1; 1 to 6; 6 to 2; 3 to 4; 4 to 5; 5 to 6; and a self-loop on vertex 5.
**Part 2.** A directed graph G has 5 vertices, numbered 1 through 5. The \(5 \times 5\) matrix \(A\) is the adjacency matrix for G. The matrices \(A^2\) and \(A^3\) are given below.
\[
A^2 = \begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
\[
A^3 = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
Use the information given to answer the questions about the graph G.
(a) Which vertices can reach vertex 2 by a walk of length 3?
(b) Is there a walk of length 4 from vertex 4 to vertex 5 in G? (Hint: \(A^4 = A^2 \cdot A^2\))](https://content.bartleby.com/qna-images/question/27f445ea-1d9a-48bd-a007-85d03e874c3c/e4e9d189-1096-4e2d-bc24-d6e7bce5b9ac/kwh6cqs_thumbnail.png)
Transcribed Image Text:**Problem 6**
**Part 1.** Give the adjacency matrix for the graph G as pictured below:
![Graph with 6 vertices and 9 edges](image.png)
*Figure 2:* A graph shows 6 vertices and 9 edges. The vertices are 1, 2, 3, 4, 5, and 6, represented by circles. The edges between the vertices are represented by arrows, as follows: 4 to 3; 3 to 2; 2 to 1; 1 to 6; 6 to 2; 3 to 4; 4 to 5; 5 to 6; and a self-loop on vertex 5.
**Part 2.** A directed graph G has 5 vertices, numbered 1 through 5. The \(5 \times 5\) matrix \(A\) is the adjacency matrix for G. The matrices \(A^2\) and \(A^3\) are given below.
\[
A^2 = \begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
\[
A^3 = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 0
\end{bmatrix}
\]
Use the information given to answer the questions about the graph G.
(a) Which vertices can reach vertex 2 by a walk of length 3?
(b) Is there a walk of length 4 from vertex 4 to vertex 5 in G? (Hint: \(A^4 = A^2 \cdot A^2\))
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