Question

In a simple regular graph, total degree is 28. If the graph has more than one cycle in it, then the degree of each vertex is

• A. 2
• B. 4
• C. 7
• D. 14

Hint:

Use handshaking lemma: sum of degrees = \(2E\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A regular graph of degree \( k \) with \( n \) vertices has a total degree of \( n \times k \). A "cycle" in graph theory refers to a path that starts and ends at the same vertex.
: Key Formula or Approach:
Total Degree \( = n \times k = 28 \).
Step 2: Detailed Explanation:
We have \( n \times k = 28 \). Possible pairs for \( (n, k) \) are:
1. \( n=14, k=2 \): If every vertex has degree 2, the graph is a cycle or collection of cycles. However, if the graph is connected and has 14 vertices of degree 2, it forms exactly one cycle (\( C_{14} \)). The question states it has "more than one cycle".
2. \( n=7, k=4 \): Every vertex has degree 4. A 4-regular graph with 7 vertices will contain multiple cycles.
3. \( n=4, k=7 \): Degree cannot exceed \( n-1 \). Here \( 7>3 \), so this is impossible.
4. \( n=2, k=14 \): Impossible as \( 14>1 \).
Given the condition of multiple cycles and simple graph constraints, the degree \( k = 4 \) is the consistent choice.
Step 3: Final Answer:
The degree of each vertex is 4.