On distance Laplacian spectral ordering of some graphs

Abstract

For a connected graph $\textsf{G}$, let $D^{L}(\textsf{G})$ be its distance Laplacian matrix ($D^{L}$ matrix) and $\rightthreetimes _{1}(\textsf{G})\ge \rightthreetimes _{2}(\textsf{G})\ge \dots \ge \rightthreetimes _{n-1}(\textsf{G})>\rightthreetimes _{n}(\textsf{G})=0$ 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 $\rightthreetimes _{n-1}$, the $D^{L}$ spectral radius $\rightthreetimes _{1}$ and the $D^{L}$ energy. Furthermore, we will give extreme values of $\rightthreetimes _1(\textsf{G})$ and of $\rightthreetimes _{n-1}(\textsf{G})$ over a class of unicyclic graphs and their complements. We present decreasing behaviour of these graphs in terms of $\rightthreetimes _1(\textsf{G}), \rightthreetimes _{n-1}(\textsf{G})$ 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.