Brouwer type conjecture for the eigenvalues of distance Laplacian matrix of a graph

The distance Laplacian matrix of a connected graph G is defined by DL(G)=Tr(G)-D(G), where TrG is the diagonal matrix with vertex transmissions of G and DG is the distance matrix of G. The distance Laplacian eigenvalues of G are denoted by ∂nLG≤∂n-1LG≤⋯≤∂1LG. For a connected graph G with order n and size m, we denote by UkG=∂1LG+⋯+∂kLG the sum of k largest distance Laplacian eigenvalues of G. In this paper, we firstly obtain a relation between the sum of the distance Laplacian eigenvalues of the graph G and the sum of the Laplacian eigenvalues of the complement G¯ of G. Then we show that graphs of diameter one and connected graphs of diameter 2 with given large maximum degree for all k satisfy Uk(G)≤W(G)+k+23, where W(G) is the transmission (or Wiener index) of G.