On the distance and distance Laplacian eigenvalues of graphs

Let G=(V,E)G=(V,E) be a simple graph with vertex set V(G)={v1,v2,…,vn}V(G)={v1,v2,…,vn} and edge set E(G)E(G). Let D(G)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 λn−k(D)≤−1λn−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 asDL(G)=Tr(G)−D(G),DL(G)=Tr(G)−D(G), where Tr(G)Tr(G) is the diagonal matrix of vertex transmissions of G. Aouchiche and Hansen (2014) [3] conjectured that m(λ1(DL))≤n−2m(λ1(DL))≤n−2 when G≇KnG≇Kn, and the equality holds if and only if either G≅K1,n−1G≅K1,n−1 or G≅Kn2,n2. In this paper, we confirm the conjecture.