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On the normalized distance laplacian eigenvalues of graphs

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

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.

Suggested Citation

  • Ganie, Hilal A. & Rather, Bilal Ahmad & Das, Kinkar Chandra, 2023. "On the normalized distance laplacian eigenvalues of graphs," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006889
    DOI: 10.1016/j.amc.2022.127615
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    References listed on IDEAS

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    1. Sun, Shaowei & Das, Kinkar Ch., 2019. "On the second largest normalized Laplacian eigenvalue of graphs," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 531-541.
    2. Huang, Xueyi & Das, Kinkar Chandra & Zhu, Shunlai, 2022. "Toughness and normalized Laplacian eigenvalues of graphs," Applied Mathematics and Computation, Elsevier, vol. 425(C).
    3. Ganie, Hilal A., 2021. "On the distance Laplacian energy ordering of a tree," Applied Mathematics and Computation, Elsevier, vol. 394(C).
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