Let's break this down step-by-step.

Problem:

You have a simple graph with 6 vertices:

• 5 vertices, each with degree 3
• 1 vertex with degree 5

Step 1: Understanding degrees and edges

The degree of a vertex in a graph is the number of edges incident to it.

To find the total number of edges in a graph, you can use the following formula: $\sum_{i=1}^n \deg(v_i) = 2E$ where:

• $\deg(v_i)$ is the degree of vertex $i$,
• $E$ is the number of edges.

The sum of all degrees is always twice the number of edges, because each edge is counted twice (once for each vertex it connects).

Step 2: Sum of degrees

Given the graph has:

• 5 vertices, each with degree 3, contributing a total degree of $5 \times 3 = 15$,
• 1 vertex with degree 5, contributing a total degree of $5$.

Thus, the total degree of all vertices is: $15 + 5 = 20$

Step 3: Solve for the number of edges

Now use the formula $\sum_{i=1}^n \deg(v_i) = 2E$: $20 = 2E$ Solving for $E$, we get: $E = \frac{20}{2} = 10$

Conclusion:

There are 10 edges in the graph.

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Would you like further details on this? Here are some follow-up questions:

1. How would this problem change if the vertex with degree 5 was replaced by a vertex with degree 4?
2. What is the maximum number of edges that can exist in a simple graph with 6 vertices?
3. How can you prove that the sum of degrees in a simple graph is always even?
4. What is the relationship between the number of edges and the degree sequence in a graph?
5. Can a simple graph with 6 vertices and 10 edges be planar?

Tip: A simple graph cannot have multiple edges between the same pair of vertices or loops (edges connecting a vertex to itself).