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On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

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Abstract

Let G be a simple connected graph with order n. Let \(\mathcal{L}(G)\) and \(\mathcal{Q}(G)\) be the normalized Laplacian and normalized signless Laplacian matrices of G, respectively. Let λk(G) be the k-th smallest normalized Laplacian eigenvalue of G. Denote by ρ(A) the spectral radius of the matrix A. In this paper, we study the behaviors of λ2(G) and \(\rho(\mathcal{L}(G))\) when the graph is perturbed by three operations. We also study the properties of \(\rho(\mathcal{L}(G))\) and X for the connected bipartite graphs, where X is a unit eigenvector of \(\mathcal{L}(G)\) corresponding to \(\rho(\mathcal{L}(G))\). Meanwhile we characterize all the simple connected graphs with \(\rho(\mathcal{L}(G))=\rho(\mathcal{Q}(G))\).

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Acknowledgments

The authors are very grateful to the referees for their detailed comments and valuable suggestions.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, 710072, China

    Xiao-guo Tian, Li-gong Wang & You Lu

  2. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, China

    Li-gong Wang

Authors
  1. Xiao-guo Tian
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  2. Li-gong Wang
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  3. You Lu
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Corresponding author

Correspondence to Li-gong Wang.

Additional information

This paper is supported by the National Natural Science Foundation of China (No. 11871398), the Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JM1032) and the Fundamental Research Funds for the Central Universities (No. 3102019ghjd003).

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Tian, Xg., Wang, Lg. & Lu, Y. On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph. Acta Math. Appl. Sin. Engl. Ser. 37, 628–644 (2021). https://doi.org/10.1007/s10255-021-1032-x

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  • DOI: https://doi.org/10.1007/s10255-021-1032-x

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