Draw nonisomorphic, undirected graphs, each having four vertices and four edges. (Select all that apply.)

To address this question, we need to draw **nonisomorphic, undirected graphs** with **four vertices and four edges**. Nonisomorphic graphs are those that cannot be transformed into each other by relabeling vertices. Here's how we can generate and differentiate the graphs:

### Step-by-step process:

1. **Determine the number of vertices and edges:**
   - We have 4 vertices labeled \(V_1, V_2, V_3, V_4\).
   - There are exactly 4 edges in each graph.

2. **Explore different configurations:**
   - The structure of these graphs must differ in terms of vertex degrees, cycles, or connections.

### Graphs to consider:

1. **Graph 1:**  
   - A **cycle of 4 vertices**:  
     Each vertex has degree 2, and they are connected in a closed loop:
     \[
     V_1 \text{---} V_2 \text{---} V_3 \text{---} V_4 \text{---} V_1
     \]

2. **Graph 2:**  
   - A **triangle (cycle of 3 vertices)** plus an extra edge:  
     Three vertices form a triangle, and the fourth vertex is connected to one of them:
     \[
     V_1 \text{---} V_2 \text{---} V_3 \text{---} V_1 \text{, and } V_4 \text{---} V_1
     \]
     In this case, vertex \(V_4\) connects to vertex \(V_1\).

3. **Graph 3:**  
   - A **path of 3 vertices** plus an extra edge connecting two non-adjacent vertices:  
     Vertices form a path, and an additional edge connects the two outer vertices:
     \[
     V_1 \text{---} V_2 \text{---} V_3 \text{---} V_4, \text{ and } V_1 \text{---} V_4
     \]
     This graph forms a cycle but is different from the first graph.

4. **Graph 4:**  
   - A **star graph with an additional edge**:  
     Three vertices are connected to a central vertex, and an extra edge connects two of the outer vertices:
     \[
     V_1 \text{---} V_2 \text{, } V_1 \text{---} V_3 \text{, } V_1 \text{---} V_4 \text{, and } V_2 \text{---} V_3
     \]
     This graph has one central vertex (degree 3) and the others (degree 2).

### Conclusion:
The four nonisomorphic graphs with 4 vertices and 4 edges are:

1. A 4-cycle.
2. A 3-cycle with one additional edge.
3. A path of length 3 with one additional edge.
4. A star-like structure with one additional edge.

Would you like to see a visual representation of these graphs? Or do you have any questions?

Here are five questions to deepen the understanding:
1. What makes two graphs nonisomorphic?
2. How would the number of edges change if we added one more vertex?
3. What is the degree sequence, and why is it important for graph comparison?
4. Can two graphs have the same number of edges and vertices but still be isomorphic?
5. How does graph connectivity influence whether graphs are nonisomorphic?

**Tip:** In graph theory, comparing degree sequences and the presence of cycles is a quick way to check if two graphs might be nonisomorphic.

Learn how to draw nonisomorphic, undirected graphs, each having four vertices and four edges. This problem explores key concepts in graph theory, such as degree sequences and nonisomorphic graph structures. Ideal for undergraduate students studying graph theory.

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