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On graphs with the largest Laplacian index

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Abstract

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.

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Authors and Affiliations

  1. Department of Mathematics, South China Normal University, Guangzhou, 510631, P.R. China

    Bolian Liu

  2. Department of Mathematics, Penn State University, Mckeesport Campus, Mckeesport, PA, 15132, USA

    Zhibo Chen

  3. Institute of Applied Mathematics, South China Agricultural University, Guangzhou, 510642, P.R. China

    Muhuo Liu

Authors
  1. Bolian Liu
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  2. Zhibo Chen
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  3. Muhuo Liu
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Corresponding author

Correspondence to Bolian Liu.

Additional information

The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006).

The work was done when Z. Chen was on sabbatical in China.

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Liu, B., Chen, Z. & Liu, M. On graphs with the largest Laplacian index. Czech Math J 58, 949–960 (2008). https://doi.org/10.1007/s10587-008-0062-3

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  • DOI: https://doi.org/10.1007/s10587-008-0062-3

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