Yellow Submarine worksheets - Free Printable

Problem Description:

The task asks us to determine the number of students in a class based on the following information:

1. Each student has exactly 5 friends within the class.
2. If any two students are not friends, they have exactly 2 mutual friends.
3. The total number of students in the class is less than 30.

We need to find the exact number of students in the class.

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Solution Approach:

This problem can be approached using concepts from graph theory. Let's represent the class as a graph where:
- Each student is a vertex.
- An edge between two vertices indicates that the corresponding students are friends.

#### Step 1: Understanding the Graph Properties
1. Each student has exactly 5 friends: This means the graph is regular with degree 5. In other words, every vertex has degree 5.
2. If two students are not friends, they have exactly 2 mutual friends: This property is characteristic of a strongly regular graph. A strongly regular graph \( G \) is defined by the parameters \( (n, k, \lambda, \mu) \), where:
- \( n \): Number of vertices (students).
- \( k \): Degree of each vertex (number of friends per student, which is 5).
- \( \lambda \): Number of common neighbors of two adjacent vertices.
- \( \mu \): Number of common neighbors of two non-adjacent vertices (which is 2).

Thus, the graph we are dealing with is a strongly regular graph with parameters \( (n, 5, \lambda, 2) \).

#### Step 2: Using Strongly Regular Graph Properties
For a strongly regular graph \( G(n, k, \lambda, \mu) \), the following relationships hold:
1. Degree condition: Each vertex has degree \( k = 5 \).
2. Adjacency condition: For any two adjacent vertices, they share \( \lambda \) common neighbors.
3. Non-adjacency condition: For any two non-adjacent vertices, they share \( \mu = 2 \) common neighbors.

Additionally, the number of vertices \( n \) must satisfy the eigenvalue condition for strongly regular graphs. However, we can also use combinatorial arguments to narrow down the possibilities.

#### Step 3: Narrowing Down the Possibilities
Given that \( n < 30 \), we need to find a value of \( n \) that satisfies the properties of a strongly regular graph with \( k = 5 \) and \( \mu = 2 \).

A well-known result in graph theory is that the Petersen graph is a strongly regular graph with parameters \( (10, 3, 0, 1) \). However, we need a graph with \( k = 5 \) and \( \mu = 2 \). One such graph is the pentagon-dodecahedron graph, which is a strongly regular graph with parameters \( (20, 5, 0, 2) \).

#### Step 4: Verifying the Parameters
The pentagon-dodecahedron graph has:
- \( n = 20 \) vertices.
- Each vertex has degree \( k = 5 \).
- Any two adjacent vertices share \( \lambda = 0 \) common neighbors (since it is a 5-regular graph with no triangles).
- Any two non-adjacent vertices share \( \mu = 2 \) common neighbors.

This matches all the given conditions:
1. Each student has exactly 5 friends.
2. If two students are not friends, they have exactly 2 mutual friends.
3. The total number of students \( n \) is less than 30.

#### Step 5: Conclusion
The number of students in the class is \( n = 20 \).

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Final Answer:

\[
\boxed{20}
\]