On the normalized distance laplacian eigenvalues of graphs

Hilal A. Ganie, Bilal Ahmad Rather and Kinkar Chandra Das

Applied Mathematics and Computation, 2023, vol. 438, issue C

Abstract: The normalized distance Laplacian matrix (DL-matrix) of a connected graph Γ is defined by DL(Γ)=I−Tr(Γ)−1/2D(Γ)Tr(Γ)−1/2, where D(Γ) is the distance matrix and Tr(Γ) is the diagonal matrix of the vertex transmissions in Γ. In this article, we present interesting spectral properties of DL(Γ)-matrix. We characterize the graphs having exactly two distinct DL-eigenvalues which in turn solves a conjecture proposed in [26]. We characterize the complete multipartite graphs with three distinct DL-eigenvalues. We present the bounds for the DL-spectral radius and the second smallest eigenvalue of DL(Γ)-matrix and identify the candidate graphs attaining them. We also identify the classes of graphs whose second smallest DL-eigenvalue is 1 and relate it with the distance spectrum of such graphs. Further, we introduce the concept of the trace norm (the normalized distance Laplacian energy DLE(Γ) of Γ) of I−DL(Γ). We obtain some bounds and characterize the corresponding extremal graphs.

Keywords: Graph; Normalized distance laplacian matrix; Energy; Diameter; Wiener index (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006889

DOI: 10.1016/j.amc.2022.127615

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