17. Let there be at most $n$ cities, with cities serving as vertices. If two cities have a direct flight, then an edge is drawn between the corresponding vertices, forming graph $G$. According to the problem, each vertex $V$ in graph $G$ has at most three adjacent vertices. The vertices adjacent to these three vertices (excluding $V$) have at most 6 more vertices. From the city corresponding to $V$, one can reach at most 9 cities directly or with one layover, plus $V$ itself, so $n \leqslant 10$. On the other hand, when $n=10$, the arrangement is as shown in Figure $35-8$, hence this country can have at most 10 cities.
17. In a certain country, an aviation network has already been established, where any city is connected to no more than three other cities, and from any city to another city, it takes no more than one transfer. How many cities can this country have at most?