Example 2: In a football invitational tournament, sixteen cities participate, each city sending Team A and Team B. According to the competition rules, each pair of teams plays at most one match, and the two teams from the same city do not play against each other. After several days of the tournament, it was found that, except for Team A of City A, all other teams have played a different number of matches. How many matches has Team B of City A played? Please prove your conclusion.
Proof: Consider 32 teams from 16 cities as vertices. If two teams have played a match, connect an edge between the two vertices, resulting in a graph $G$. By property 1: Among these 32 vertices, there are at least two vertices with the same degree. According to the given conditions: Except for the vertex representing Team A from City A, the degrees of the other 31 vertices are all different, hence they are: $0, 1, 2, 3, \cdots, 30$. Furthermore, by the given conditions: The vertices with degrees 30 and 0 represent teams from the same city, the vertices with degrees 29 and 1 represent teams from the same city, $\cdots \cdots$, the vertices with degrees 16 and 14 represent teams from the same city. Therefore, only the vertices representing Team A and Team B from City A have a degree of 15. Hence, Team B from City A has played only 15 matches.