Uniqueness of the extremal graph in the problem of maximizing the number of independent sets in regular graphs

TL;DR

This paper proves the unique extremal structure of regular graphs that maximize the number of independent sets, addressing a key combinatorial optimization problem in graph theory.

Contribution

It establishes the uniqueness of the extremal regular graph maximizing independent sets for given parameters, a previously unresolved problem.

Findings

Proves the extremal graph is unique under specified conditions.

Confirms the maximum number of independent sets is attained by a specific regular graph.

Addresses a fundamental question in extremal combinatorics.

Abstract

The main purpose of this paper is to prove the uniqueness of a graph attaining the maximum of the number of independent sets over all $k$-regular graphs on $n$ vertices for $2k|n$.