2. Let $G$ be a simple graph of order 100. It is known that for any vertex $u$, there exists another vertex $v$ such that $u$ and $v$ are adjacent, and there is no vertex adjacent to both $u$ and $v$. Find the maximum possible number of edges in graph $G$. (Provided by Yunhao Fu)
2. Let $G=(V, E)$.

For $u v \in E$, if there is no other vertex adjacent to both $u$ and $v$, then $u v$ is called a "good edge".
Let $E_{0}$ be the set of all good edges, and $G_{0}=\left(V, E_{0}\right)$.
From the problem statement, we know that each vertex in graph $G$ has at least one good edge, i.e., $G_{0}$ has no isolated vertices.

Perform the following operation on graph $G_{0}$: For an edge $u v \in E_{0}$ in graph $G_{0}$, if the degrees of both $u$ and $v$ in $G_{0}$ are at least 2, delete this edge, and the resulting graph still has no isolated vertices. Continue this operation until a graph $G_{1}=\left(V, E_{1}\right)$ is obtained, which has no isolated vertices, and for any edge $u v \in E_{1}$, at least one of the vertices $u$ or $v$ has a degree of 1 in $G_{1}$.

First, consider a connected component of graph $G_{1}$, with $n$ vertices.
When $n=2$, it is $K_{1,1}$.
Assume $n \geqslant 3$. Let $u v$ be one of the edges, and the degree of $u$ is at least 2. Suppose $u$ is adjacent to $v_{1}, v_{2}, \cdots, v_{k}$. Then, $v_{1}, v_{2}, \cdots, v_{k}$ have a degree of 1 in $G_{1}$, and this connected component is a star graph $K_{1, k}(k=n-1)$. Combining the definition of good edges, we know that $v_{1}, v_{2}, \cdots, v_{n-1}$ are not adjacent in $G$. Thus, $G_{1}$ is the union of several star graphs.
Let $G_{1}$ have $m$ connected components, with $n_{1}, n_{2}, \cdots, n_{m}$ vertices respectively. Then, $n_{i} \geqslant 2, \sum_{i=1}^{m} n_{i}=100$.
Next, consider the number of edges between two connected components $K_{1, n_{i}-1}$ and $K_{1, n_{j}-1}$ in $G$.

Let the center of the first star graph be $u$, and the pendant vertices be $u_{1}, u_{2}, \cdots, u_{n_{i}-1}$; the center of the second star graph be $v$, and the pendant vertices be $v_{1}, v_{2}, \cdots, v_{n_{i}-1}$. The following is considered in $G$.

If $u v \in E$, then $u_{1}, u_{2}, \cdots, u_{n_{i}-1}$ are not adjacent to $v$, and $v_{1}, v_{2}, \cdots, v_{n_{j}-1}$ are not adjacent to $u$. Therefore, the number of edges between these two connected components is at most $\left(n_{i}-1\right)\left(n_{j}-1\right)+1$. If $u$ and $v$ are not adjacent, suppose $u$ is adjacent to $s$ vertices among $v_{1}, v_{2}, \cdots, v_{n_{j}-1}$, denoted as $v_{1}, v_{2}, \cdots, v_{s}$. Suppose $v$ is adjacent to $t$ vertices among $u_{1}, u_{2}, \cdots, u_{n_{i}-1}$, denoted as $u_{1}, u_{2}, \cdots, u_{t}$. Then, $v_{1}, v_{2}, \cdots, v_{s}$ are not adjacent to any $u_{k}\left(1 \leqslant k \leqslant n_{i}-1\right)$, and $u_{1}, u_{2}, \cdots, u_{t}$ are not adjacent to any $v_{k}(1 \leqslant k \leqslant n_{j}-1)$. Therefore, the number of edges between these two connected components is at most
$$
\begin{array}{l}
\left(n_{i}-t-1\right)\left(n_{j}-s-1\right)+t+s \\
\leqslant\left(n_{i}-1\right)\left(n_{j}-1\right)+1 .
\end{array}
$$

Then $|E|$
$$
\begin{array}{l}
\leqslant \sum_{i=1}^{m}\left(n_{i}-1\right)+\sum_{1 \leqslant i1200 .
\end{array}
$$

When $17 \leqslant m \leqslant 50$,

Thus $f(m)<3800$
For $1 \leqslant m \leqslant 16$, calculate sequentially, we know
$m \neq 8$ when, $f(m) \leqslant 3822 ; f(8)=3823$.
If $|E|=f(8)=3823$, the equality holds when $n_{1}=n_{2}=\cdots=n_{8}$, but 100 is not divisible by 8, so the equality does not hold.
Therefore, $|E| \leqslant 3822$.
Finally, construct a 100-vertex graph $G$: First, take eight disjoint star graphs, four of which are $K_{1,11}$, and the other four are $K_{1,12}$. Connect the center points of any two star graphs, and connect each non-center point of the first star graph to each non-center point of the second star graph. Then
$$
\begin{array}{l}
n_{1}=n_{2}=n_{3}=n_{4}=12, \\
n_{5}=n_{6}=n_{7}=n_{8}=13, \\
|E|=\sum_{i=1}^{8}\left(n_{i}-1\right)+\sum_{1 \leqslant i<j \in 8}\left(\left(n_{i}-1\right)\left(n_{j}-1\right)+1\right) \\
=3822 .
\end{array}
$$

Note that the edges in the initial star graphs are all good edges. Therefore, each vertex has a good edge, and graph $G$ meets the requirements.
In summary, the maximum number of edges in graph $G$ is 3822.