What is the maximum number of edges in a bipartite graph with n even vertices ?

Key idea: If the two parts of a bipartite graph have sizes a and b with a + b = n, then the maximum possible number of edges is a × b (every vertex in one part connected to every vertex in the other).

Step 1: Express the edge count in one parameter:

• Let one part have a vertices, so the other has n - a vertices. The number of edges is a × (n - a).

Step 2: Maximize the product a(n - a).

• The product a(n - a) is a quadratic in a with maximum at a = n/2. Therefore the product is maximized when the two parts are as equal as possible.

Conclusion: For even n take a = n/2, giving maximum edges = (n/2) × (n/2) = n^2/4.