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!