## Abstract

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.

Original language | English |
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Pages (from-to) | 867-892 |

Number of pages | 26 |

Journal | Journal of Applied Mathematics and Computing |

Volume | 70 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2024 |

## Keywords

- 05C12
- 05C50
- 15A18
- Distance Laplacian energy
- Distance Laplacian matrix
- Double star
- Laplacian matrix
- Spectral invariant ordering

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics