An upper bound for the number of independent sets in regular graphs

TL;DR

This paper improves the upper bound on the number of independent sets in regular graphs, approaching a conjectured tight bound, by establishing a new upper bound on the independent set polynomial.

Contribution

It introduces a new upper bound on the independent set polynomial for regular graphs, matching the conjectured bound in the leading terms of the exponent.

Findings

Improved upper bound on independent set count in regular graphs.

New bound on the independent set polynomial $P(\lambda,G)$.

Enhanced bounds on the number of independent sets of fixed size.

Abstract

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If true, this bound would be tight, being achieved by the disjoint union of $N/2d$ copies of $K_{d,d}$. Kahn established the bound for bipartite $G$, and later gave an argument that established $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{2}{d}\right)} $$ for $G$ not necessarily bipartite. In this note, we improve this to $$ i(G)\leq 2^{\frac{N}{2}\left(1+\frac{1+o(1)}{d}\right)} $$ where $o(1) \rightarrow 0$ as $d \rightarrow \infty$, which matches the conjectured upper bound in the first two terms of the exponent. We obtain this bound as a corollary of a new upper bound on the independent set polynomial $P(\lambda,G)=\sum_{I \in {\cal I}(G)} \lambda^{|I|}$ of an $N$-vertex, $d$-regular graph $G$, namely $$ P(\gl,G) \leq (1+\gl)^{\frac{N}{2}} 2^{\frac{N(1+o(1))}{2d}} $$ valid for all $\gl > 0$. This also allows us to improve the bounds obtained recently by Carroll, Galvin and Tetali on the number of independent sets of a fixed size in a regular graph.