Question 13

Let $V_1$ and $V_2$ be the two disjoint independent sets of vertices in the bipartite graph, with $|V_1| = n_1$ and $|V_2| = n_2$.

Given total vertices $n_1 + n_2 = 12$.

The maximum possible number of edges occurs in the complete bipartite graph $K_{n_1, n_2}$, where every vertex in $V_1$ is connected to every vertex in $V_2$:

For a fixed sum $n_1 + n_2 = 12$, the product $n_1 n_2$ is maximized when the two parts are as equal as possible:

The maximum number of edges is: