The problem describes a $(7,10)$ graph $G$ with 7 vertices and 10 edges. The degree of a vertex in a graph refers to the number of edges connected to it. The graph has 6 vertices of degree $a$ and 1 vertex of degree $b$, and we are asked to determine the value of $b$.

Step 1: Use the Handshaking Lemma

The Handshaking Lemma states that the sum of the degrees of all vertices in a graph is twice the number of edges. In other words:

$\sum_{v \in V} \deg(v) = 2 \times \text{(number of edges)}$

Here, the number of edges is 10, so:

$\sum_{v \in V} \deg(v) = 2 \times 10 = 20$

Step 2: Express the total degree in terms of $a$ and $b$

The graph has 6 vertices of degree $a$ and 1 vertex of degree $b$. The total sum of degrees is:

$6a + b = 20$

Step 3: Determine possible values for $a$

Since the degree of a vertex must be an integer and the maximum possible degree for any vertex in a simple graph with 7 vertices is 6 (since each vertex can be connected to up to 6 other vertices), we know $a$ must be an integer between 1 and 6.

Step 4: Solve for $b$

Let's try different values of $a$ to find $b$.

• If $a = 3$:

$6(3) + b = 20$ $18 + b = 20 \quad \Rightarrow \quad b = 2$

Thus, $a = 3$ and $b = 2$ satisfy the equation.

Conclusion:

The value of $b$ is $2$.

Would you like more details on how to apply this to other types of graphs, or do you have any further questions?

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Here are 5 related questions:

1. What is the Handshaking Lemma, and how is it applied in different graph types?
2. How can you determine the maximum degree of a vertex in a simple graph?
3. What is the relationship between the number of vertices and edges in a regular graph?
4. How would the degrees change if additional edges were added to the graph?
5. Can you generalize the approach to solving for degrees in a weighted or directed graph?

Tip: The degree of a vertex is always at least 1 (unless isolated), and no vertex can have a degree greater than $n-1$, where $n$ is the total number of vertices.