2. With the above as vertices, for any two people who have shaken hands, connect an edge between the corresponding two vertices, to form a graph $G$. Since not every two people have shaken hands, the graph $G$ has the maximum number of edges when it is a subgraph $K_{N}-e$, where $e$ is the line connecting the two people $A_{1}$ and $A_{2}$ who have not shaken hands, so $N-2$ is the answer.
2. (1978 US Olympiad Question) In a room, there are $N(73)$ people, and at least one person has not shaken hands with everyone in the room. What is the maximum number of people with whom someone in the room could have shaken hands?