On distance Laplacian spectral ordering of some graphs

• Published in: Journal of applied mathematics & computing Vol. 70; no. 1; pp. 867 - 892
• Main Authors: Rather, Bilal Ahmad, Aouchiche, Mustapha, Imran, Muhammad, El Hallaoui, Issmail
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2024; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Eigenvalues; Eigenvectors; Extreme values; Graphs; Invariants; Mathematical analysis; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Spectrum analysis; Theory of Computation; Distance Laplacian matrix; 05C12; Distance Laplacian energy; Spectral invariant ordering; 05C50; Double star; Laplacian matrix; 15A18
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-024-01995-8

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.