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

Let Gbe a connected graph of order nand m(G)(I) be the number of Laplacian eigenvalues of Gin an interval I. If I= {lambda} for a real number lambda, then m(G)(lambda) is just the multiplicity of lambda as a Laplacian eigenvalue of G. It is well known that the Laplacian eigenvalues of Gare 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 1in [0, n]. Particularly, Hedetniemi etal. (2016) [14] proved that mG[0, 1) =.if Ghas domination number lambda. 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 m(G)(n -1, n] = lambda and m(G)(n-1, n] = gamma - 1, where lambda and lambda are respectively the vertex-connectivity and the chromatic number of G. Two other main results of this paper focus on mG(lambda), the multiplicity of an arbitrary Laplacian eigenvalue.of G. It is proved that m(G)(lambda) = n - m(G)(lambda) = <= Delta/Delta+ 1 and for a connected graph Gwith domination number lambda and maximum degree Delta. (C) 2020 Elsevier Inc. All rights reserved.