A new eigenvalue bound for independent sets

Let G be a simple, undirected, and connected graph on n vertices with eigenvalues λ1 ≤ ... ≤ λn. Moreover, let m, δ, and α denote the size, the minimum degree, and the independence number of G, respectively. W.H. Haemers proved α ≤ −λ1λn δ2−λ1λnn and, if η is the largest Laplacian eigenvalue of G, then α ≤ η−δ η n is shown by C.D. Godsil and M.W. Newman. We prove α ≤ 2σ−2 σδ m for the largest normalized eigenvalue σ of G, if δ ≥ 1. For ε > 0, an infinite family Fε of graphs is constructed such that 2σ−2 σδ m = α < ( 3 + ε)min{ −λ1λn δ2−λ1λnn, η−δ η n} for all G ∈ Fε. Moreover, a sequence of graphs is presented showing that the difference between 2σ−2 σδ m and D.M. Cvetković’s upper bound min{|{i ∈ {1, ..., n}|λi ≤ 0}|, |{i ∈ {1, ..., n}|λi ≥ 0}|} on α can be arbitrarily small.