On the maximum number of maximum independent sets

TL;DR

This paper presents simplified proofs and bounds on the maximum number of maximum independent sets in graphs and trees, extending classical theorems and characterizing extremal structures.

Contribution

It offers new bounds and characterizations for the maximum number of maximum independent sets in graphs and trees, generalizing and simplifying previous results.

Findings

Bound on maximum independent sets in general graphs based on Turán's theorem

Exact bounds for trees with given order and independence number

Bounds for subcubic trees involving Fibonacci-related exponential functions

Abstract

We give a very short and simple proof of Zykov's generalization of Tur\'{a}n's theorem, which implies that the number of maximum independent sets of a graph of order $n$ and independence number $\alpha$ with $\alpha<n$ is at most $\left\lceil\frac{n}{\alpha}\right\rceil^{n\,{\rm mod}\,\alpha} \left\lfloor\frac{n}{\alpha}\right\rfloor^{\alpha-(n\,{\rm mod}\,\alpha)}$. Generalizing a result of Zito, we show that the number of maximum independent sets of a tree of order $n$ and independence number $\alpha$ is at most $2^{n-\alpha-1}+1$, if $2\alpha=n$, and, $2^{n-\alpha-1}$, if $2\alpha>n$, and we also characterize the extremal graphs. Finally, we show that the number of maximum independent sets of a subcubic tree of order $n$ and independence number $\alpha$ is at most $\left(\frac{1+\sqrt{5}}{2}\right)^{2n-3\alpha+1}$, and we provide more precise results for extremal values of $\alpha$.