Problem 1.
In the following, let
V = {0, 1, 2, 3, 4, 5, 6},
E = {(0, 2), (3, 2), (5, 6), (6, 1), (3, 0)}.
For this problem, you do not need to show your work.
a) Draw the undirected graph G = (V, E). Remember that when writing the edges of an undirected
graph, we often abuse notation and write (u, v) when we really mean {u, v}; we have done so here.
b) Draw the graph G = (V, E), assuming that G is directed.
c) Write down the connected components of G, assuming that G is undirected.
d) Write the adjacency matrix representation of G, assuming that G is undirected.
e) Write the adjacency matrix representation of G, assuming that G is directed

Let's tackle the problem step by step:

### Part (a): Draw the Undirected Graph \( G = (V, E) \)
Given:
- **Vertices**: \( V = \{0, 1, 2, 3, 4, 5, 6\} \)
- **Edges**: \( E = \{(0, 2), (3, 2), (5, 6), (6, 1), (3, 0)\} \)

For the undirected graph, each edge \( (u, v) \) means that there is a connection between vertices \( u \) and \( v \) in both directions. Here is the graph representation:

- Vertex 0 is connected to vertices 2 and 3.
- Vertex 1 is connected to vertex 6.
- Vertex 2 is connected to vertices 0 and 3.
- Vertex 3 is connected to vertices 0 and 2.
- Vertex 5 is connected to vertex 6.
- Vertex 6 is connected to vertices 5 and 1.

### Part (b): Draw the Directed Graph \( G = (V, E) \)
For the directed graph, each edge \( (u, v) \) represents a directed connection from vertex \( u \) to vertex \( v \). Here's the directed graph:

- Edge from 0 to 2.
- Edge from 3 to 2.
- Edge from 5 to 6.
- Edge from 6 to 1.
- Edge from 3 to 0.

### Part (c): Connected Components of \( G \) (Undirected)
For the undirected graph, connected components are subsets of vertices such that each vertex in the subset is reachable from any other vertex in the subset.

The connected components are:
- **Component 1**: \( \{0, 2, 3\} \) (connected via edges \( (0, 2), (3, 2), (3, 0) \))
- **Component 2**: \( \{1, 5, 6\} \) (connected via edges \( (5, 6), (6, 1) \))
- **Component 3**: \( \{4\} \) (vertex 4 is isolated)

### Part (d): Adjacency Matrix Representation of \( G \) (Undirected)
For the undirected graph, the adjacency matrix \( A \) is symmetric, with \( A[i][j] = 1 \) if there is an edge between vertex \( i \) and vertex \( j \), and \( A[i][j] = 0 \) otherwise.

\[
A = \begin{pmatrix}
0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}
\]

### Part (e): Adjacency Matrix Representation of \( G \) (Directed)
For the directed graph, the adjacency matrix \( A \) is not necessarily symmetric, with \( A[i][j] = 1 \) if there is a directed edge from vertex \( i \) to vertex \( j \), and \( A[i][j] = 0 \) otherwise.

\[
A = \begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 & 0 \\
\end{pmatrix}
\]

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Would you like more details on any part of this problem or have other questions?

Here are 5 related questions to consider:
1. How does the adjacency matrix change if a new edge is added to the graph?
2. How would the adjacency matrix differ if the graph were directed and weighted?
3. What is the significance of the diagonal elements in the adjacency matrix for an undirected graph?
4. How can the adjacency matrix be used to determine if the graph is connected?
5. What is the interpretation of higher powers of the adjacency matrix in terms of path lengths?

**Tip:** When analyzing graphs, visualizing both the graph and its adjacency matrix can provide deeper insights into the structure and connectivity of the graph.

Explore a detailed solution to a graph theory problem involving undirected and directed graphs, connected components, and adjacency matrix representations. Understand the structure and connectivity of graphs using specific examples.

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