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 $mis(G)=\mathrm{\mid }MIS(G)\mathrm{\mid }$. In this paper, we show that for any tree $T$ with $n$ vertices and independence number $\alpha$, $$mis(T)\ge f(n-\alpha ),$$ 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\ $\text{and}$\ α=2, 3, & $\text{if}$ ; α=n-2 ; $\text{and}$ ; n\neq4, 2f(n-α), & $\text{if}$ ; n\geq 5; $\text{and}$; \lceil \frac{n}{2} \rceil \leq α< n-2, f(n-α+2)-f(n-α-3), &$\text{if}$ ; n\geq 5, ;$\text{and}$\ n ; $\text{is odd}$, ; $\text{and}$ ; α= \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.