Distribution of Laplacian eigenvalues of graphs

Let G be a graph of order n with m edges and clique number ω  . Let μ1≥μ2≥…≥μn=0μ1≥μ2≥…≥μn=0 be the Laplacian eigenvalues of G   and let σ=σ(G)σ=σ(G)(1≤σ≤n)(1≤σ≤n) be the largest positive integer such that μσ≥2mn. In this paper we study the relation between σ and ω. In particular, we provide the answer to Problem 2.3 raised in Pirzada and Ganie (2015) [15]. Moreover, we characterize all connected threshold graphs with σ<ω−1σ<ω−1, σ=ω−1σ=ω−1 and σ>ω−1σ>ω−1. We obtain Nordhaus–Gaddum-type results for σ. Some relations between σ with other graph invariants are obtained.