bipartite graph with vertex partition?

Question

Let $G$ be a bipartite graph with vertex partition $V(G) = V_1 \cup V_2$ such that $|V_1|= a$ and $|V_2| = a + 2$. Show that: $|E(G)| \leq a^2 + 2a$

Answer 1

Let us say that $G$ is a bipartite graph with partitions $V_1$ and $V_2$, such that $|V_1| = a$ and $|V_2| = a + 2$. Because the graph is bipartite, we know that every edge connects one vertex from $V_1$ to a vertex from $V_2$. For each vertex in $V_1$, it can be connected to, at most, $a + 2$ vertices since that is the number of vertices in $V_2$. Given that there are $a$ vertices in $V_1$, we can have a maximum of $a(a + 2) = a^2 + 2a$ edges in the whole graph.