A graph has a 3 x 3 adjacency matrix with 2's in every cell on the descending diagonal and 1's in all other cells. How many walks of length two are possible on the graph? The descending diagonal for the 3 x 3 matrix A is comprised of the cells a1,1, a2,2, a3,3.
A graph has a 3 x 3 adjacency matrix with 2's in every cell on the descending diagonal and 1's in all other cells. How many walks of length two are possible on the graph? The descending diagonal for the 3 x 3 matrix A is comprised of the cells a1,1, a2,2, a3,3.
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Author:Erwin Kreyszig
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![### Understanding Walks in a Graph Using Adjacency Matrix
**Problem Statement:**
A graph has a 3x3 adjacency matrix with 2's in every cell on the descending diagonal and 1's in all other cells. How many walks of length two are possible on the graph?
The descending diagonal for the 3x3 matrix \( A \) is comprised of the cells \( a_{1,1}, a_{2,2}, a_{3,3} \).
**Explanation:**
In this problem, we are dealing with an adjacency matrix representing a graph. The entries in the matrix indicate the likelihood or capacity for moving from one vertex to another. Specifically:
- The values on the descending diagonal (i.e., from the top left to the bottom right) are 2.
- The rest of the values are 1.
#### Adjacency Matrix \( A \):
Let's illustrate the given 3x3 adjacency matrix \( A \):
\[
A = \begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\]
Here:
- \( a_{1,1} = 2 \)
- \( a_{2,2} = 2 \)
- \( a_{3,3} = 2 \)
- All other entries \( a_{ij} \) (where \( i \neq j \)) are 1.
#### Walks of Length Two:
A walk of length two involves moving from a starting vertex to an intermediate vertex and then to the ending vertex. Mathematically, the number of such walks can be found by squaring the adjacency matrix \( A \):
\[
A^2 = A \times A
\]
For the matrix \( A \), the product \( A^2 \) will provide the number of two-step walks between each pair of vertices. Let's compute \( A^2 \):
\[
A^2 = \begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\times
\begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\]
Calculating the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b3df428-2a91-49dd-84ab-611ffd39f8ca%2F64d9f122-4283-4c3d-8593-ec3e33520de6%2Fvfzmpe7_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Walks in a Graph Using Adjacency Matrix
**Problem Statement:**
A graph has a 3x3 adjacency matrix with 2's in every cell on the descending diagonal and 1's in all other cells. How many walks of length two are possible on the graph?
The descending diagonal for the 3x3 matrix \( A \) is comprised of the cells \( a_{1,1}, a_{2,2}, a_{3,3} \).
**Explanation:**
In this problem, we are dealing with an adjacency matrix representing a graph. The entries in the matrix indicate the likelihood or capacity for moving from one vertex to another. Specifically:
- The values on the descending diagonal (i.e., from the top left to the bottom right) are 2.
- The rest of the values are 1.
#### Adjacency Matrix \( A \):
Let's illustrate the given 3x3 adjacency matrix \( A \):
\[
A = \begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\]
Here:
- \( a_{1,1} = 2 \)
- \( a_{2,2} = 2 \)
- \( a_{3,3} = 2 \)
- All other entries \( a_{ij} \) (where \( i \neq j \)) are 1.
#### Walks of Length Two:
A walk of length two involves moving from a starting vertex to an intermediate vertex and then to the ending vertex. Mathematically, the number of such walks can be found by squaring the adjacency matrix \( A \):
\[
A^2 = A \times A
\]
For the matrix \( A \), the product \( A^2 \) will provide the number of two-step walks between each pair of vertices. Let's compute \( A^2 \):
\[
A^2 = \begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\times
\begin{pmatrix}
2 & 1 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2 \\
\end{pmatrix}
\]
Calculating the
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