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On the second largest normalized Laplacian eigenvalue of graphs

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  • Sun, Shaowei
  • Das, Kinkar Ch.

Abstract

Let G=(V,E) be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. The normalized Laplacian spread of graph G, denoted by ρ1−ρn−1, is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ2 of graphs. Moreover, we give a lower bound on ρ2 of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:531-541
    DOI: 10.1016/j.amc.2018.12.023
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    References listed on IDEAS

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    1. Li, Jianxi & Guo, Ji-Ming & Shiu, Wai Chee & Altındağ, Ş. Burcu Bozkurt & Bozkurt, Durmuş, 2018. "Bounding the sum of powers of normalized Laplacian eigenvalues of a graph," Applied Mathematics and Computation, Elsevier, vol. 324(C), pages 82-92.
    2. Huang, Jing & Li, Shuchao, 2018. "The normalized Laplacians on both k-triangle graph and k-quadrilateral graph with their applications," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 213-225.
    3. Xie, Pinchen & Zhang, Zhongzhi & Comellas, Francesc, 2016. "The normalized Laplacian spectrum of subdivisions of a graph," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 250-256.
    4. Xie, Pinchen & Zhang, Zhongzhi & Comellas, Francesc, 2016. "On the spectrum of the normalized Laplacian of iterated triangulations of graphs," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 1123-1129.
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    Cited by:

    1. 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).

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