a graph contains 21 edges, 3 vertices of degree 4 and all other vertices of degree 2. find the total number of vertices

To solve the problem, let's break it down step by step.

### Given Information:
- The graph has 21 edges.
- 3 vertices have a degree of 4.
- All other vertices have a degree of 2.

### Step 1: Use the Degree Sum Formula
The sum of the degrees of all vertices in a graph is equal to twice the number of edges. This is known as the **Handshaking Lemma**:
\[
\sum \text{{degree of all vertices}} = 2 \times \text{{number of edges}}
\]

Given:
- Total number of edges = 21

Thus,
\[
\sum \text{{degree of all vertices}} = 2 \times 21 = 42
\]

### Step 2: Calculate the Contribution from the Vertices of Degree 4
There are 3 vertices of degree 4, so their contribution to the degree sum is:
\[
3 \times 4 = 12
\]

### Step 3: Calculate the Contribution from the Other Vertices
Let \( n \) be the total number of vertices, and let \( x \) be the number of vertices with degree 2. Since the total number of vertices includes the 3 vertices of degree 4:
\[
n = x + 3
\]

The contribution from the vertices with degree 2 is:
\[
x \times 2
\]

### Step 4: Set Up the Equation
The sum of the degrees from all vertices should equal 42:
\[
12 + 2x = 42
\]

### Step 5: Solve for \( x \)
Subtract 12 from both sides:
\[
2x = 30
\]

Divide by 2:
\[
x = 15
\]

### Step 6: Calculate the Total Number of Vertices
The total number of vertices is:
\[
n = x + 3 = 15 + 3 = 18
\]

### Final Answer:
The total number of vertices in the graph is **18**.

Would you like more details or have any questions? Here are some related problems:

1. How does the degree-sum formula apply to complete graphs?
2. What is the minimum number of vertices in a graph where all vertices have even degrees?
3. How do the degrees of vertices relate to the number of connected components in a graph?
4. How many edges would a complete graph with 18 vertices have?
5. Can a graph have exactly two vertices of odd degree? Why or why not?

**Tip:** In a simple graph, the sum of the degrees of all vertices is always even because each edge contributes exactly 2 to the total degree.

Explore how to calculate the total number of vertices in a graph with 21 edges, where 3 vertices have a degree of 4 and all other vertices have a degree of 2. Understand the application of the Handshaking Lemma and solve step-by-step. Suitable for advanced high school students.

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