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

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?

---

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.

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

Learn how to determine the degree of a vertex in a (7,10) graph using the Handshaking Lemma. This step-by-step solution explains how to solve for b when 6 vertices have a degree of a and 1 vertex has a degree of b, focusing on key graph theory concepts. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation