Proof for four conjectures about the distance Laplacian and distance signless Laplacian eigenvalues of a graph

The distance Laplacian matrix L(G)L(G) of a graph G   is defined to be L(G)=diag(Tr)−D(G)L(G)=diag(Tr)−D(G), where D(G)D(G) denotes the distance matrix of G   and diag(Tr)diag(Tr) denotes the diagonal matrix of the vertex transmissions in G. Similarly, the distance signless Laplacian matrix of G   is defined as Q(G)=diag(Tr)+D(G)Q(G)=diag(Tr)+D(G). The eigenvalues of L(G)L(G) and Q(G)Q(G) are called the distance Laplacian and distance signless Laplacian eigenvalues, respectively. In this paper, four conjectures proposed by M. Aouchche and P. Hansen about the largest and the second largest distance Laplacian eigenvalues and the second largest distance signless Laplacian eigenvalue of a graph are proved.