Example 4. (85 Provincial Six Autonomous Regions High School Mathematics Joint Competition Question) A football invitational tournament involves sixteen cities, each city sending Team A and Team B. According to the competition rules, each pair of teams plays at most one match, and teams from the same city do not play against each other. After several days of the tournament, a statistical review found that, except for Team A from City A, all other teams have played a different number of matches. How many matches has Team B from City A played? Please prove your conclusion.
Prove that if two teams have played against each other, then an edge is connected between the corresponding two vertices to form graph $G$. Let the team from City A be $v^{*}$, then the original problem is equivalent to: find the degree of the vertex corresponding to the other team from City A.
Obviously, $\max d\left(v_{i}\right) \leqslant 32-2=30$, i.e., $v_{1} \in V$
$V-\left\{u^{*}\right\}$ contains 31 vertices whose degrees can only take 0, 1, 2, ..., 31 values. Therefore, the degrees of the vertices in $V-\left\{v^{*}\right\}$ are exactly 0, 1, 2, ..., 30. Let the vertex with degree $i$ be $v_{i} (i=0,1,2, \cdots, 30)$, then $V=\left\{v_{0}, v_{1}, v_{2}, \cdots, v_{30}, v^{*}\right\}$. Clearly, the only vertex not adjacent to $v_{30}$ is $v_{0}$, so $v_{30}$ and $v_{0}$ are the two teams from the same city; the only vertices not adjacent to $v_{29}$ are $v_{0}$ and $v_{1}$, so $v_{29}$ and $v_{1}$ are the two teams from the same city; ..., and the set of vertices not adjacent to $v_{16}$ is $\left\{v_{0}, v_{1}, v_{2}, \cdots, v_{14}\right\}$, so $v_{16}$ and $v_{14}$ are the two teams from the same city.

Thus, the team from the same city as $U^{\prime \prime}$ can only be $v_{15}$, so the desired number of matches is 15.

Let a graph with $n$ vertices, where each pair of vertices is connected by exactly one edge, be denoted as $k_{n}$. Clearly, the number of edges in $k_{0}$ is $\frac{1}{2} n(n-1)$.