Let us solve the given questions based on the directed graph shown in the image.

Solution:

(a) Find the indegree of each vertex.

The indegree of a vertex in a directed graph is the number of edges directed into that vertex. Let's calculate for each vertex:

1. Vertex 1: Edges directed into vertex 1: $6 \to 1$, $2 \to 1$.
   Indegree = 2

2. Vertex 2: Edges directed into vertex 2: $1 \to 2$, $4 \to 2$.
   Indegree = 2

3. Vertex 3: Edges directed into vertex 3: $2 \to 3$, $5 \to 3$.
   Indegree = 2

4. Vertex 4: Edges directed into vertex 4: $3 \to 4$, $5 \to 4$.
   Indegree = 2

5. Vertex 5: Edges directed into vertex 5: $6 \to 5$.
   Indegree = 1

6. Vertex 6: Edges directed into vertex 6: $5 \to 6$.
   Indegree = 1

(b) Construct an adjacency list.

The adjacency list represents the outgoing edges for each vertex. Here's the adjacency list:

1. Vertex 1: $1 \to 2$.
   Adjacency list: [2]

2. Vertex 2: $2 \to 1$, $2 \to 3$.
   Adjacency list: [1, 3]

3. Vertex 3: $3 \to 4$.
   Adjacency list: [4]

4. Vertex 4: $4 \to 2$.
   Adjacency list: [2]

5. Vertex 5: $5 \to 3$, $5 \to 4$, $5 \to 6$.
   Adjacency list: [3, 4, 6]

6. Vertex 6: $6 \to 1$, $6 \to 5$.
   Adjacency list: [1, 5]

Final Answers:

1. Indegree of each vertex:

   • Vertex 1: 2
   • Vertex 2: 2
   • Vertex 3: 2
   • Vertex 4: 2
   • Vertex 5: 1
   • Vertex 6: 1

2. Adjacency list:

   • $1: [2]$
   • $2: [1, 3]$
   • $3: [4]$
   • $4: [2]$
   • $5: [3, 4, 6]$
   • $6: [1, 5]$

Would you like a detailed explanation or visualization of this? Let me know! 😊

Follow-up Questions:

1. What is the difference between indegree and outdegree in a directed graph?
2. How would the adjacency matrix look for this graph?
3. Can you determine the outdegree of each vertex from the adjacency list?
4. How can the adjacency list be used to traverse the graph efficiently?
5. What is the practical application of adjacency lists in computer algorithms?

Tip:

In graph problems, always double-check edge directions carefully, as errors can often arise from misinterpreting arrows!