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.