TY - JOUR
AU - Pirzada, Shariefuddin
AU - Khan, Saleem
PY - 2023/02/01
Y2 - 2023/03/25
TI - On the sum of distance Laplacian eigenvalues of graphs
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 54
IS - 1
SE - Papers
DO - 10.5556/j.tkjm.54.2023.4120
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4120
SP - 83-91
AB - <p>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 $G$.</p>
ER -