A bipartite graph is one where the vertices can be divided into two distinct sets such that no two vertices within the same set are adjacent (i.e., there are no edges connecting vertices in the same set).

Let’s analyze each graph:

• Graph A: The vertices can be divided into two sets: {a, c, e, g} and {b, d, f, h}. All edges are between vertices in these two sets, so Graph A is bipartite.

• Graph B: Upon inspection, we can see that there is no possible way to divide the vertices into two sets without some vertices in the same set being adjacent. For example, if we place {a, b, c} in one set, the edges between them show that it cannot be bipartite. Thus, Graph B is not bipartite.

• Graph C: Similarly, the connections in this graph also prevent it from being divided into two distinct sets without edges within the same set. There are triangles formed in the graph, making it impossible for bipartiteness. Therefore, Graph C is not bipartite.

• Graph D: This graph appears to be a complete graph where every vertex is connected to every other vertex. Such graphs are never bipartite because you can't divide the vertices into two sets without creating edges within one of the sets. Thus, Graph D is not bipartite.

Conclusion:

The only graph that is bipartite is Graph A.

Would you like further details or explanations? Here are some related questions for more exploration:

1. What are the conditions for a graph to be bipartite?
2. How can you detect if a graph is bipartite using a breadth-first search (BFS)?
3. Are there any real-world applications of bipartite graphs?
4. What is the relationship between bipartite graphs and even-length cycles?
5. How does the chromatic number relate to bipartite graphs?

Tip: A simple way to check if a graph is bipartite is by trying to color the graph using two colors—if possible, the graph is bipartite.