Vertex-connectivity, chromatic number, domination number, maximum degree and Laplacian eigenvalue distribution

Abstract

Let G be a connected graph of order n and mG(I) be the number of Laplacian eigenvalues of G in an interval I. If I={λ} for a real number λ, then mG(λ) is just the multiplicity of λ as a Laplacian eigenvalue of G. It is well known that the Laplacian eigenvalues of G are all in the interval [0,n]. A lot of attention has been paid to the distribution of Laplacian eigenvalues in the smallest subinterval [0,1) of length 1 in [0,n]. Particularly, Hedetniemi et al. (2016) [14] proved that mG[0,1)≤γ if G has domination number γ. We are interested in another extreme problem: The distribution of Laplacian eigenvalues in the largest subinterval (n−1,n] of length 1. In this article, we prove that mG(n−1,n]≤κ and mG(n−1,n]≤χ−1, where κ and χ are respectively the vertex-connectivity and the chromatic number of G. Two other main results of this paper focus on mG(λ), the multiplicity of an arbitrary Laplacian eigenvalue λ of G. It is proved that mG(λ)≤n−γ and mG(λ)≤ΔΔ+1n for a connected graph G with domination number γ and maximum degree Δ.