Open AccessOn the Sum of Distance Laplacian Eigenvalues of Graphs
Author(s) -
S. Pirzada,
Saleem Khan
Publication year - 2021
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.54.2023.4120
Subject(s) - mathematics , combinatorics , resistance distance , eigenvalues and eigenvectors , laplacian matrix , distance matrix , laplace operator , vertex (graph theory) , diagonal matrix , diagonal , connectivity , algebraic connectivity , order (exchange) , graph , matrix (chemical analysis) , upper and lower bounds , wiener index , discrete mathematics , mathematical analysis , graph power , geometry , physics , line graph , materials science , finance , quantum mechanics , economics , composite material
Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.