prove that The maximum degree of any vertex in a simple graph with n vertices is n−1.

To prove that the maximum degree of any vertex in a simple graph with \( n \) vertices is \( n - 1 \), let's analyze the conditions carefully.

### Step-by-Step Proof

1. **Definition of Vertex Degree:**
   - The degree of a vertex \( v \) in a graph \( G \) is defined as the number of edges incident to \( v \). In other words, it's the number of vertices that \( v \) is directly connected to.

2. **Simple Graph Definition:**
   - A simple graph is one with no loops (edges that connect a vertex to itself) and no multiple edges (more than one edge between any two vertices).

3. **Upper Bound on Vertex Degree:**
   - In a simple graph with \( n \) vertices, any vertex \( v \) can connect to at most \( n - 1 \) other vertices (all the other vertices in the graph).
   - Therefore, the degree of any vertex \( v \) cannot exceed \( n - 1 \).

4. **Achievability of \( n - 1 \) Degree:**
   - It’s possible for a vertex to have degree \( n - 1 \) if the graph is complete. A complete graph on \( n \) vertices, denoted \( K_n \), has every pair of vertices connected by an edge.
   - In \( K_n \), each vertex is connected to every other vertex, so each vertex indeed has degree \( n - 1 \).

5. **Conclusion:**
   - Thus, in a simple graph with \( n \) vertices, the maximum degree of any vertex is \( n - 1 \), and this bound is achieved in a complete graph.

### Summary

The maximum degree of any vertex in a simple graph with \( n \) vertices is \( n - 1 \), which is achievable in the complete graph \( K_n \).

Would you like more details or have any questions?

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Here are some related questions to explore:

1. What is the minimum degree of any vertex in a simple connected graph with \( n \) vertices?
2. How does the degree sequence of a simple graph change as vertices are added?
3. How many edges are in a complete graph with \( n \) vertices?
4. What is the degree of each vertex in a bipartite complete graph \( K_{m, n} \)?
5. How does removing an edge affect the degree of a vertex?

**Tip:** In graph theory, complete graphs are valuable examples for understanding upper bounds on properties like vertex degree and edge count.

Prove that the maximum degree of any vertex in a simple graph with n vertices is n−1.

Explore the proof of the maximum degree of any vertex in a simple graph with n vertices, demonstrating that it cannot exceed n - 1. This solution covers concepts in graph theory, including vertex degree and properties of complete graphs, making it suitable for high school or college-level students.

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