Construct a graph based on the adjacency matrix that appears below. Label all nodes with indices consistent with the placement of numbers within the matrix.
⌈0 6 0 5 0⌉
| 6 0 1 0 3 |
| 0 1 0 4 8 |
| 5 0 4 0 0 |
⌊0 3 8 0 0⌋
Describe the graph and why it is consistent with the matrix.
How many simple paths are there from vertex 1 to vertex 5? Explain.Which is the shortest of those paths?
To construct the graph based on the given adjacency matrix, we can represent each number in the matrix as a weight on the edge between two nodes. Nodes can be labeled according to the indices consistent with the placement of numbers within the matrix.
The adjacency matrix:
⌈0 6 0 5 0⌉
| 6 0 1 0 3 |
| 0 1 0 4 8 |
| 5 0 4 0 0 |
⌊0 3 8 0 0⌋
Based on the matrix, we have 5 nodes represented by their indices, as follows:
Node 1 is connected to Node 2 with a weight of 6 and Node 4 with a weight of 5.
Node 2 is connected to Node 1 with a weight of 6, Node 3 with a weight of 1, and Node 5 with a weight of 3.
Node 3 is connected to Node 2 with a weight of 1, Node 4 with a weight of 4, and Node 5 with a weight of 8.
Node 4 is connected to Node 1 with a weight of 5 and Node 3 with a weight of 4.
Node 5 is connected to Node 2 with a weight of 3 and Node 3 with a weight of 8.
The graph can be represented as:
6 ----------- 2 ----------- 5
/ \ / \ / \
1 5 1 4 3 8
| | | | | |
1 | 3 | 2 |
\ / \ / \ /
4 ----------- 3 ----------- 1
This graph is consistent with the given adjacency matrix because it shows the connections and weights of the edges as represented in the matrix.
To find the number of simple paths from vertex 1 to vertex 5, we can analyze the graph. In this case, there are 3 simple paths from vertex 1 to vertex 5:
Path 1: 1 -> 2 -> 5
Path 2: 1 -> 4 -> 3 -> 5
Path 3: 1 -> 2 -> 3 -> 5
The shortest path among these is Path 1: 1 -> 2 -> 5. The total weight of this path is 6 + 3 = 9, which is the smallest compared to the other paths.
To construct a graph based on the given adjacency matrix, we can convert the non-zero elements in the matrix into edges. Each row and column in the matrix corresponds to a specific vertex in the graph.
The vertices of the graph can be labeled as follows:
Vertex 1 corresponds to the first row and column.
Vertex 2 corresponds to the second row and column.
Vertex 3 corresponds to the third row and column.
Vertex 4 corresponds to the fourth row and column.
Vertex 5 corresponds to the fifth row and column.
Now, we can identify the edges based on the non-zero elements in the matrix. Starting from the top-left element (vertex 1), we can see that there is an edge from vertex 1 to vertex 2 with weight 6, an edge from vertex 1 to vertex 4 with weight 5, and no edge from vertex 1 to vertex 3 or vertex 5.
Continuing in the same way, we can determine the remaining edges:
- Vertex 2 is connected to vertex 1 with weight 6, vertex 3 with weight 1, and vertex 5 with weight 3.
- Vertex 3 is connected to vertex 2 with weight 1, vertex 4 with weight 4, and vertex 5 with weight 8.
- Vertex 4 is connected to vertex 1 with weight 5 and vertex 3 with weight 4.
- Vertex 5 is connected to vertex 2 with weight 3 and vertex 3 with weight 8.
Now, let's describe the graph:
- Vertex 1 is connected to vertex 2 with weight 6 and vertex 4 with weight 5.
- Vertex 2 is connected to vertex 1 with weight 6, vertex 3 with weight 1, and vertex 5 with weight 3.
- Vertex 3 is connected to vertex 2 with weight 1, vertex 4 with weight 4, and vertex 5 with weight 8.
- Vertex 4 is connected to vertex 1 with weight 5 and vertex 3 with weight 4.
- Vertex 5 is connected to vertex 2 with weight 3 and vertex 3 with weight 8.
The graph is consistent with the given matrix because the non-zero elements of the matrix represent the weights of the edges connecting the corresponding vertices in the graph.
To determine the number of simple paths from vertex 1 to vertex 5, we can explore all possible paths and count the ones that do not pass through a vertex more than once.
In this case, there are 2 simple paths from vertex 1 to vertex 5:
1. Path 1: 1 → 4 → 3 → 5.
2. Path 2: 1 → 2 → 5.
The shortest path from vertex 1 to vertex 5 is Path 2: 1 → 2 → 5.