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

TL;DR

This paper establishes lower bounds on the number of maximal independent sets in trees and unicyclic graphs based on their order and independence number, demonstrating the bounds are tight.

Contribution

It provides new sharp lower bounds for the count of maximal independent sets in specific graph classes, linking these bounds to Fibonacci numbers.

Findings

Lower bounds for trees depend on Fibonacci numbers.

Unicyclic graphs have specific minimal counts of maximal independent sets.

The bounds are proven to be sharp and tight.

Abstract

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.