On distance Laplacian spectral ordering of some graphs

For a connected graph G, let D^L(G) be its distance Laplacian matrix (D^L matrix) and ⋌₁(G)≥⋌₂(G)≥⋯≥⋌_n-1(G)>⋌_n(G)=0 be its eigenvalues. In this article, we will study the D^L spectral invariants of graphs whose complements are trees. In particular, with the technique of eigenvalue/eigenvector analysis and intermediate value theorem, we order tree complements as a decreasing sequence on the basis of their second smallest D^L eigenvalue ⋌_n-1, the D^L spectral radius ⋌₁ and the D^L energy. Furthermore, we will give extreme values of ⋌₁(G) and of ⋌_n-1(G) over a class of unicyclic graphs and their complements. We present decreasing behaviour of these graphs in terms of ⋌₁(G),⋌_n-1(G) and D^L energy. Thereby, we obtain complete characterization of graphs minimizing/maximizing with respect to there spectral invariants over class of these unicyclic graphs.