On the Sum of Distance Laplacian Eigenvalues of Graphs

Authors

  • Shariefuddin Pirzada University of Kashmir
  • Saleem Khan University of Kashmir

DOI:

https://doi.org/10.5556/j.tkjm.54.2023.4120

Keywords:

Distance matrix, distance Laplacian matrix, distance Laplacian eigenvalue, diameter, Wiener index

Abstract

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 $G$.

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Published

2021-11-07

How to Cite

Pirzada, S., & Khan, S. (2021). On the Sum of Distance Laplacian Eigenvalues of Graphs. Tamkang Journal of Mathematics, 54. https://doi.org/10.5556/j.tkjm.54.2023.4120

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Papers