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

Research output: Contribution to journalArticle

2 Citations (Scopus)

### 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 Kd,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 a new bound on the number of independent sets in triangle-free cubic graphs.

Original language English 793-800 8 Discrete Mathematics 341 3 https://doi.org/10.1016/j.disc.2017.11.016 Published - 1 Mar 2018

### Fingerprint

Triangle-free Graph
Independent Set
Regular Graph
Polynomials
Polynomial
Cubic Graph
Random variables
Linear programming
Disjoint
Random variable
Lower bound
Independence

### Keywords

• Hard-core model
• Independent sets
• Occupancy fraction

### Cite this

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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 Kd,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 a new bound on the number of independent sets in triangle-free cubic graphs.",
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In: Discrete Mathematics, Vol. 341, No. 3, 01.03.2018, p. 793-800.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Minimizing the number of independent sets in triangle-free regular graphs

AU - Cutler, Jonathan

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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 Kd,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 a new bound on the number of independent sets in triangle-free cubic graphs.

AB - 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 Kd,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 a new bound on the number of independent sets in triangle-free cubic graphs.

KW - Hard-core model

KW - Independent sets

KW - Occupancy fraction

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