Research PaperNo Access

On distance Laplacian spectrum (energy) of graphs

Hilal A. Ganie

Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India

E-mail Address: hilahmad1119kt@gmail.com

https://doi.org/10.1142/S1793830920500615Cited by:16 (Source: Crossref)

Abstract

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues $ρ_{1}^{L} \geq ρ_{2}^{L} \geq \dots \geq ρ_{n}^{L}$ , the distance Laplacian energy $DLE (G)$ is defined as $DLE (G) = \sum_{i = 1}^{n} | ρ_{i}^{L} - \frac{2 W (G)}{n} |$ , where $W (G)$ is the Wiener index of $G$ . We obtain the distance Laplacian spectrum of the joined union of graphs $G_{1}, G_{2}, \dots, G_{n}$ in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number $χ$ . We show that among all connected graphs with chromatic number $χ$ the complete $χ$ -partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree $\frac{2 W (G)}{n}$ .

Keywords:

AMSC: 05C50, 05C12, 15A18

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