Cardinalities of the total number of independent sets

TL;DR

This paper investigates the range of possible counts of independent sets in n-vertex graphs, showing it is very close to 2^n with specific bounds, and explores related combinatorial problems.

Contribution

It establishes bounds on the cardinality of the set of possible independent set counts and characterizes this set precisely at x=1 for n-vertex graphs.

Findings

The set of possible independent set counts has size close to 2^n.

Bounds on the ratio of this set size to 2^n are established.

Connections to additive combinatorial problems are demonstrated.

Abstract

We study the set of numbers the total number of independent sets can admit in $n$-vertex graphs. In this paper, we prove that the cardinality $\mathcal{N}i(n)$ of this set is very close to $2^n$ in the following sense: $\mathcal{N}i(n)/2^n = O(n^{-1/5})$ while for infinitely many $n$, we have $\log_2(\mathcal{N}i(n)/2^n)\ge -2^{(1+o(1)\sqrt{\log_2 n}}$. This set is also precisely the set of possible values of the independence polynomial $I_G(x)$ at $x=1$ for $n$-vertex graphs $G$. As an application, we address an additive combinatorial problem on subsets of a given vector space that avoid certain intersection patterns with respect to subspaces.