This problem asks about a complete undirected graph with five vertices. In a complete graph, every vertex is connected to every other vertex by an edge.

Step-by-step solution:

1. Vertices (n): There are 5 vertices in this graph.

2. Edges: In a complete graph, the number of edges is given by the formula: $\text{Number of edges} = \frac{n(n - 1)}{2}$ where $n$ is the number of vertices.

3. Substitute $n = 5$: $\text{Number of edges} = \frac{5(5 - 1)}{2} = \frac{5 \times 4}{2} = 10$

Thus, the complete graph on five vertices has 10 edges.

Would you like a detailed explanation of any part of this problem?

Here are 5 related questions to further explore the topic:

1. What is the general structure of a complete graph with $n$ vertices?
2. How many edges would a complete graph with 6 vertices have?
3. How do you differentiate between a complete graph and other types of graphs?
4. Can a directed graph also be complete, and how would the edge count differ?
5. What are real-world examples of applications of complete graphs in networks?

Tip:

In graph theory, the complete graph on $n$ vertices is often denoted by $K_n$, where $n$ represents the number of vertices.