A (7,10) graph G has 6 vertices of degree a and 1 of degree b l. What is the value of b? Explain

To solve this, we need to use the **Handshaking Lemma**, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges.

### Step-by-Step Solution

1. **Graph Information**:
   - The graph \( G \) has 7 vertices and 10 edges, so it is a \( (7,10) \) graph.
   - 6 of the vertices have degree \( a \).
   - 1 vertex has degree \( b \), and we are tasked with finding \( b \).

2. **Handshaking Lemma**:
   The sum of the degrees of all vertices is equal to twice the number of edges:
   \[
   \sum \text{degrees of vertices} = 2 \times \text{number of edges} = 2 \times 10 = 20
   \]

3. **Expressing the sum of degrees**:
   - There are 6 vertices with degree \( a \), so the total degree contribution from these vertices is \( 6a \).
   - There is 1 vertex with degree \( b \), so its degree contribution is \( b \).

   Therefore, the total degree sum can be written as:
   \[
   6a + b = 20
   \]

4. **Solving for \( b \)**:
   We also know that the sum of degrees must satisfy the above equation. Therefore:
   \[
   b = 20 - 6a
   \]

   So, the value of \( b \) depends on the degree \( a \) of the 6 vertices.

Without additional constraints on \( a \), the value of \( b \) is \( 20 - 6a \), and the answer will depend on the specific degree \( a \).

### Conclusion
The value of \( b \) is \( 20 - 6a \). To find the exact value of \( b \), the degree \( a \) of the 6 vertices must be known or provided.

Let me know if you have any questions, or if you'd like more details about how to choose the degree \( a \)!

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**Relative Questions:**
1. How is the Handshaking Lemma applied in graph theory?
2. What are other properties of a \( (7,10) \) graph?
3. Can the degrees of vertices in a graph ever be negative? Why or why not?
4. What would happen to the value of \( b \) if \( a \) were larger or smaller?
5. How would this problem change for a different number of vertices or edges?

**Tip:** Always check if your graph is simple (no loops or multiple edges) before applying the Handshaking Lemma!

Explore how to find the degree of a vertex in a (7,10) graph using the Handshaking Lemma. This problem involves 6 vertices with degree a and 1 vertex with degree b, helping students understand the relationship between vertex degrees and graph edges. Suitable for Grades 9-12.

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