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

Abstract

The distance Laplacian matrix of a connected graph G is defined by ${D^L}(G)=Tr(G)-D(G)$, where $Tr\left( G\right)$ is the diagonal matrix with vertex transmissions of G and $D\left( G\right)$ is the distance matrix of G. The distance Laplacian eigenvalues of G are denoted by ${\partial _n^L\left( G\right) }\le {\partial _{n-1}^L\left( G\right) }\le \cdots \le {\partial _1^L\left( G\right) }$. For a connected graph G with order n and size m, we denote by ${U_k}\left( G\right) =\partial _1^L\left( G\right) +\cdots +\partial _k^L\left( G\right)$ 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 $\overline{G}$ 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 $U_k(G) \le W(G)+\left( {\begin{array}{c}k+2\\ 3\end{array}}\right) ,$ where W(G) is the transmission (or Wiener index) of G.