Combinatorics Graph Theory Proof problem I am struggling with 9.31 from A Walk Through Combinatorics by Miklos Bona. The problem statement reads:

There are several people in a classroom; some of them know each other. It is true that if two people know the same number of people in the classroom, then there is nobody in the classroom both these people know. Prove that there is someone in the classroom who knows exactly one other person in the classroom.

I realized that the second sentence of this means that if two vertices have the same degree, the sets of their neighbors are disjoint. I am not sure how to proceed past this point, however. Any help or direction would be appreciated.

Hint. I guess "some of them know each other" means that $\Delta(G)\gt0$ where $\Delta(G)$ is the maximum degree. Let $v$ be a vertex of degree $\Delta(G)$. Let $w_1,w_2,\dots,w_{\Delta(G)}$ be the neighbors of $v$; what can you say about their degrees?
