Distribution of Laplacian eigenvalues of graphs

Let G be a graph of order n with m edges and clique number ω. Let μ₁≥μ₂≥…≥μ_n=0 be the Laplacian eigenvalues of G and let σ=σ(G) (1≤σ≤n) be the largest positive integer such that μ_σ≥[formula presented]. 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−1, σ=ω−1 and σ>ω−1. We obtain Nordhaus–Gaddum-type results for σ. Some relations between σ with other graph invariants are obtained.