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The image shows a question related to graph theory. There are three parts to the question:

1. **List the neighbors of vertex \( a \)**:
   - In graph theory, the neighbors of a vertex are the vertices directly connected to it by an edge.
   - In this graph, vertex \( a \) is connected to vertices \( b \), \( c \), and \( d \).
   - So, the neighbors of vertex \( a \) are: \( b, c, d \).

2. **What is the degree of vertex \( a \)?**:
   - The degree of a vertex is the number of edges incident to it (i.e., how many edges are connected to the vertex).
   - Vertex \( a \) has three edges (one each to \( b \), \( c \), and \( d \)).
   - Thus, the degree of vertex \( a \) is \( 3 \).

3. **What is the total degree of the graph?**:
   - The total degree of the graph is the sum of the degrees of all the vertices.
   - If we consider the vertices:
     - Vertex \( a \) has degree \( 3 \) (as discussed).
     - Vertex \( b \) has degree \( 1 \) (one edge connecting to \( a \)).
     - Vertex \( c \) has degree \( 1 \) (one edge connecting to \( a \)).
     - Vertex \( d \) has degree \( 1 \) (one edge connecting to \( a \)).
   - The total degree is: \( 3 + 1 + 1 + 1 = 6 \).

Let me know if you would like more details or have further questions!

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Here are 5 related questions for further exploration:
1. How do you determine if a graph is connected?
2. What is a complete graph and how does its degree compare to this one?
3. What is the difference between degree and total degree in a graph?
4. How would you find the degree sequence of a graph?
5. Can you identify any isolated vertices in a graph?

**Tip:** The sum of the degrees of all vertices in a graph is always twice the number of edges, since each edge contributes to the degree of two vertices.

List the neighbors of vertex a. What is the degree of vertex a? What is the total degree of the graph?

This example demonstrates how to find the neighbors of vertex a, calculate its degree, and determine the total degree of a graph. It explores basic graph theory concepts such as vertex neighbors, degree, and the handshaking lemma. Suitable for high school students.

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