1. **Problem Statement:** Find the diameter of each graph type $N_n$, $K_n$, $P_n$, $C_n$, and $W_n$ and explain what $N$, $C$, $P$, and $W$ stand for. 2. **Definitions of Graphs:** - $N_n$: The null graph with $n$ vertices and no edges. - $K_n$: The complete graph with $n$ vertices where every pair of vertices is connected. - $P_n$: The path graph with $n$ vertices connected in a single line. - $C_n$: The cycle graph with $n$ vertices connected in a closed loop. - $W_n$: The wheel graph with $n$ vertices formed by connecting a single central vertex to all vertices of a cycle $C_{n-1}$. 3. **Diameter Explanation:** The diameter of a graph is the greatest distance between any pair of vertices, where distance is the length of the shortest path connecting them. 4. **Diameters:** - For $N_n$, since there are no edges, no paths exist between distinct vertices, so diameter is undefined or infinite: $$\text{diameter}(N_n) = \infty$$ - For $K_n$, every vertex is directly connected to every other vertex, so the longest shortest path is 1 edge: $$\text{diameter}(K_n) = 1$$ - For $P_n$, the longest shortest path is between the two end vertices of the path, which is $n-1$ edges: $$\text{diameter}(P_n) = n - 1$$ - For $C_n$, the longest shortest path is half the cycle length, rounded down: $$\text{diameter}(C_n) = \left\lfloor \frac{n}{2} \right\rfloor$$ - For $W_n$, the central vertex connects to all others, so the longest shortest path is 2 edges (between two non-central vertices via the center): $$\text{diameter}(W_n) = 2$$ 5. **Summary of what letters stand for:** - $N$: Null graph (no edges) - $K$: Complete graph - $P$: Path graph - $C$: Cycle graph - $W$: Wheel graph **Final answers:** $$\text{diameter}(N_n) = \infty$$ $$\text{diameter}(K_n) = 1$$ $$\text{diameter}(P_n) = n - 1$$ $$\text{diameter}(C_n) = \left\lfloor \frac{n}{2} \right\rfloor$$ $$\text{diameter}(W_n) = 2$$