Answer: B. $n_1 + n_2 + p_1p_2$
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Key Concept: Graph Join ($G_1 \vee G_2$)
The join $G_1 \vee G_2$ of two disjoint graphs is formed by:
1. Taking the union of $G_1$ and $G_2$
2. Adding all possible edges between vertices of $G_1$ and $G_2$
Edge Count:
$$|E(G_1 \vee G_2)| = \underbrace{n_1}_{\text{edges in } G_1} + \underbrace{n_2}_{\text{edges in } G_2} + \underbrace{p_1 \cdot p_2}_{\text{cross edges}}$$
Why $p_1 \cdot p_2$ cross edges?
- Every vertex in $G_1$ (there are $p_1$ of them) connects to every vertex in $G_2$ (there are $p_2$ of them)
- So new edges added $= p_1 \times p_2$
Final Answer:
$$\boxed{n_1 + n_2 + p_1p_2}$$
