On the distance and distance Laplacian eigenvalues of graphs

Let G = (V, E) be a simple graph with vertex set V (G) = {v(1), v(2),...,v(n)} and edge set E(G). Let D(G) be the distance matrix of G. For a given nonnegative integer k, when n is sufficiently large with respect to k, we show that lambda(n-k) (D) <= -1, thereby solving a problem proposed by Lin et al. (2014) [8]. The distance Laplacian spectral radius of a connected graph G is the spectral radius of the distance Laplacian matrix of G, defined as D-L(G) = Tr(G) - D(G), where Tr(G) is the diagonal matrix of vertex transmissions of G. Aouchiche and Hansen (2014) [3] conjectured that m(lambda(1)(D-L)) <= n - 2 when G not congruent to K-n, and the equality holds if and only if either G not congruent to K-1,K-n-1 or G = K-n/2,K-n/2. In this paper, we confirm the conjecture. (C) 2015 Elsevier Inc. All rights reserved.