Minimizing the number of independent sets in triangle-free regular graphs

TL;DR

This paper uses linear programming methods to establish lower bounds on the independence polynomial and the number of independent sets in triangle-free regular graphs, extending previous results on maximizing independence polynomials.

Contribution

It introduces a novel application of linear programming bounds to derive lower bounds for independence polynomials in regular graphs, especially triangle-free ones.

Findings

Lower bounds on independence polynomial for regular graphs

New bounds on independent sets in triangle-free regular graphs

Extension of existing results to a broader class of graphs

Abstract

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give new bounds on the number of independent sets in triangle-free regular graphs.