To find the minimum number of edges in an undirected graph $G$ with 40 vertices, where the degree of each vertex is at least 5, we use the Handshaking Lemma and properties of graphs.

Step 1: Handshaking Lemma

The Handshaking Lemma states: $\sum_{v \in V} \deg(v) = 2|E|$ where $|E|$ is the number of edges in the graph.

Step 2: Lower Bound on Total Degree

If the degree of each vertex is at least 5, the total degree of the graph is at least: $\text{Total Degree} \geq 40 \times 5 = 200.$

Step 3: Solve for Minimum Number of Edges

Using the Handshaking Lemma: $2|E| \geq 200.$ Divide through by 2: $|E| \geq 100.$

Step 4: Verify Feasibility

The minimum number of edges $|E| = 100$ is possible if each vertex has exactly degree 5. Such a graph is called a 5-regular graph. It is feasible to construct a 5-regular graph with 40 vertices since:

• Each vertex has degree 5.
• The sum of all degrees is $40 \times 5 = 200$, which is even (a requirement for simple graphs).

Thus, the minimum number of edges is: $\boxed{100}.$

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Related Questions:

1. What is the maximum number of edges that an undirected graph with 40 vertices can have?
2. How do you construct a 5-regular graph with 40 vertices?
3. What is the relationship between vertex degrees and the graph's connectivity?
4. How does the minimum number of edges change if the minimum degree is increased to 6?
5. What are some applications of regular graphs in computer science and mathematics?

Tip:

The degree of a vertex is closely tied to edge counts and connectivity. Always verify feasibility by considering total degree constraints and the graph's structure.