On distance Laplacian spectral ordering of some graphs

For a connected graph \(\textsf{G}\) $\mathsf{G}$ , let \( D^{L}(\textsf{G}) \) $D^L(\mathsf{G})$ be its distance Laplacian matrix ( \(D^{L}\) $D^L$ matrix) and \( \rightthreetimes _{1}(\textsf{G})\ge \rightthreetimes _{2}(\textsf{G})\ge \dots \ge \rightthreetimes _{n-1}(\textsf{G})>\rightthreetimes _{n}(\textsf{G})=0 \) ${⋌}_1(\mathsf{G})\ge {⋌}_2(\mathsf{G})\ge \cdots \ge {⋌}_{n-1}(\mathsf{G})>{⋌}_n(\mathsf{G})=0$ be its eigenvalues. In this article, we will study the \(D^{L}\) $D^L$ 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 \(D^{L}\) $D^L$ eigenvalue \( \rightthreetimes _{n-1} \) ${⋌}_{n-1}$ , the \(D^{L}\) $D^L$ spectral radius \(\rightthreetimes _{1}\) ${⋌}_1$ and the \(D^{L}\) $D^L$ energy. Furthermore, we will give extreme values of \(\rightthreetimes _1(\textsf{G})\) ${⋌}_1(\mathsf{G})$ and of \(\rightthreetimes _{n-1}(\textsf{G})\) ${⋌}_{n-1}(\mathsf{G})$ over a class of unicyclic graphs and their complements. We present decreasing behaviour of these graphs in terms of \(\rightthreetimes _1(\textsf{G}), \rightthreetimes _{n-1}(\textsf{G})\) ${⋌}_1(\mathsf{G}),{⋌}_{n-1}(\mathsf{G})$ and \(D^{L}\) $D^L$ energy. Thereby, we obtain complete characterization of graphs minimizing/maximizing with respect to there spectral invariants over class of these unicyclic graphs.