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 A and A are given below. 010 0 0 0 0100 100 0 0 A 100 10 101 1000 0 0100 0 0 0100 01101 A 1 1010 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?
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 A and A are given below. 010 0 0 0 0100 100 0 0 A 100 10 101 1000 0 0100 0 0 0100 01101 A 1 1010 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?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**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.
\[ A^2 = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix} \]
\[ A^3 = \begin{pmatrix}
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1
\end{pmatrix} \]
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%2F9f2920b9-16cb-409a-b0d9-184574689b7b%2F860671a2-e237-4729-ac11-0cb0e41b8a8d%2F6fsk5uu_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
\[ A^2 = \begin{pmatrix}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 1 & 0
\end{pmatrix} \]
\[ A^3 = \begin{pmatrix}
1 & 0 & 1 & 0 & 0 \\
0 & 1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 1
\end{pmatrix} \]
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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