On the maximum number of maximum independent sets in connected graphs

TL;DR

This paper characterizes the connected graphs with fixed order and independence number that maximize the number of maximum independent sets, confirming a conjecture and identifying a unique extremal graph structure.

Contribution

It provides a complete characterization of the extremal connected graphs maximizing maximum independent sets for given parameters, confirming a prior conjecture.

Findings

Identifies a unique extremal graph structure for given parameters.

Confirms a conjecture by Derikvand and Oboudi.

Provides a precise construction of the extremal graphs.

Abstract

We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint union of $\alpha$ cliques of orders $\left\lceil\frac{n}{\alpha}\right\rceil$ and $\left\lfloor\frac{n}{\alpha}\right\rfloor$, by selecting a vertex $x$ in a largest clique and adding an edge between $x$ and a vertex in each of the remaining $\alpha-1$ cliques. Our result confirms a conjecture of Derikvand and Oboudi [On the number of maximum independent sets of graphs, Transactions on Combinatorics 3 (2014) 29-36].