Part 1. Give the adjacency matrix for the graph G as pictured below: Figure 2: A graph shows 6 vertices and 9 edges. The vertices are 1, 2, 3, 4, 5, and 6, represented by circles. The cdges between the vertices are represented by arrous, as follows: 4 to 3; 3 to 2; 2 to i; 1 to 6; 6 to 2; 3 to 4; 4 to 5; 5 to 6; and a self loop on vertez 5. Part 2. A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. The matrices A² and Aº are given below. '0 1 0 0 0 0 0 100 10 0 00 1001 0 01101 0 0 00 0 10 0 0 0 0 10 0 0 1 1 0 1 1 1 0 1 0 Use the information given to answer the questions about the graph G. (n) 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 A² · A².)
Part 1. Give the adjacency matrix for the graph G as pictured below: Figure 2: A graph shows 6 vertices and 9 edges. The vertices are 1, 2, 3, 4, 5, and 6, represented by circles. The cdges between the vertices are represented by arrous, as follows: 4 to 3; 3 to 2; 2 to i; 1 to 6; 6 to 2; 3 to 4; 4 to 5; 5 to 6; and a self loop on vertez 5. Part 2. A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. The matrices A² and Aº are given below. '0 1 0 0 0 0 0 100 10 0 00 1001 0 01101 0 0 00 0 10 0 0 0 0 10 0 0 1 1 0 1 1 1 0 1 0 Use the information given to answer the questions about the graph G. (n) 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 A² · A².)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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part 1 and 2
![**Problem 6**
**Part 1.** Give the adjacency matrix for the graph \( G \) as pictured below:
*Image Description:*
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.
\[
A^2 =
\begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 \\
\end{bmatrix}
\]
\[
A^3 =
\begin{bmatrix}
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 & 0 \\
\end{bmatrix}
\]
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: \( A^4 = A^2 \cdot A^2 \)).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F471cb0b1-24a7-4f47-a754-0c03ff5b6852%2F680c84b3-7950-4f6a-b419-ab84cc2595b4%2F3uuyfm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6**
**Part 1.** Give the adjacency matrix for the graph \( G \) as pictured below:
*Image Description:*
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.
\[
A^2 =
\begin{bmatrix}
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 0 \\
\end{bmatrix}
\]
\[
A^3 =
\begin{bmatrix}
0 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 & 0 \\
\end{bmatrix}
\]
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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