Independent sets in quasi-regular graphs

New upper bounds for the number of independent sets in graphs are obtained. c © 2006 Published by Elsevier Ltd 1. Definitions and the statement of results All graphs under consideration are finite, undirected, and simple. The vertices are considered as numbered. Denote the degree of a vertex v by σ(v). G is called an (n, k, θ)-graph if it has n vertices and k ≤ σ(v) ≤ k + θ for any vertex v. Everywhere we assume that n and k are sufficiently large and θ = o(k) as k → ∞. A subset of vertices of a graph G is called independent if the subgraph of G induced by A does not contain an edge. The family of all independent sets of G will be denoted by I(G). We put I (G) = |I(G)|. Let G = (V ; E) be a graph with the vertex set V and the edge set E , and v ∈ V . We call the set ∂v = {u : (u, v) ∈ E} the boundary of v. It is clear that σ(v) = |∂v|. The boundary of A ⊆ V in a graph G = (V ; E) is the set ∂A = (⋃v∈A ∂v) \ A. Suppose 0 ≤ < 1. Suppose l ≤ k − θ ≤ k + θ ≤ m. A graph on n vertices with minimal degree l and maximal degree m, with ∆ as the fraction of vertices with degree more than k + θ and with δ as the fraction of vertices with degree less than k − θ , will be called a (n, l, k,m, δ,∆, θ)-graph. Such a graph is called quasi-regular if θ k and k → ∞ when k → ∞. Here, we give a brief survey on the topic and prove the following theorems. Theorem 1. Let G = (V , E) be a (n, l, k,m, δ,∆, θ)-graph. Then I (G) ≤ 2 2 ( 1+δ(1−l/k)+∆(m/k−1)+O((θ+√k log k)/k)) . (1) Denote by Iβ(G) the family of sets A ∈ I(G) with ||A| − n/4| ≥ βn/4 and Iβ(G) = |Iβ(G)|. 0195-6698/$ see front matter c © 2006 Published by Elsevier Ltd doi:10.1016/j.ejc.2006.06.017 A.A. Sapozhenko / European Journal of Combinatorics 27 (2006) 1206–121