Proposition 2 Using $m(m \geqslant 2)$ different colors to color the $n+1(n \geqslant 2)$ interconnected wheel-shaped regions $A_{0}, A_{1}$, $\cdots, A_{n}$, and any two adjacent regions must be colored differently. Then the number of different coloring schemes is
$$
a_{n}=m\left[(-1)^{n}(m-2)+(m-2)^{n}\right] .
$$
Prove: As shown in Figure 3, let the number of valid coloring schemes be $a_{n}$, then region $A_{0}$ has $m$ coloring options, and the other $n$ regions must be different from $A_{0}$, leaving only $m-1$ colors to choose from. According to Proposition 1, there are $(-1)^{n}(m-2)+(m-2)^{n}$ coloring options. Therefore,
$$
a_{n}=m\left[(-1)^{n}(m-2)+(m-2)^{n}\right] .
$$