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?

---

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.