The minimum number of maximal independent sets in graphs with given order and independence number

Let $MIS(G)$ be the set of all maximal independent sets in a graph $G$, and let $mis(G)=|MIS(G)|$. In this paper, we show that for any tree $T$ with $n$ vertices and independence number $\alpha$, $$mis(T)\geq f(n-\alpha),$$ and for any unicyclic graph $G$ with $n$ vertices and independence number $\alpha$, \begin{align*} mis(G)\geq \begin{cases} 2, & \text{if} \ n=4\ \text{and}\ \alpha=2, 3, & \text{if} \; \alpha=n-2 \; \text{and} \; n\neq4, 2f(n-\alpha), & \text{if} \; n\geq 5\; \text{and}\; \lceil \frac{n}{2} \rceil \leq \alpha < n-2, f(n-\alpha+2)-f(n-\alpha-3), &\text{if} \; n\geq 5, \;\text{and}\ n \; \text{is odd}, \; \text{and} \; \alpha = \lfloor \frac{n}{2} \rfloor, \end{cases} \end{align*} where $f(n)$ represent the $n$th Fibonacci number. Moreover, we also show that the above inequalities are sharp.