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Laplacian Spectral Radius and Maximum Degree of Trees with Perfect Matchings

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Abstract

Let μ(T) and Δ(T) denote the Laplacian spectral radius and the maximum degree of a tree T, respectively. Denote by \({\mathcal{T}_{2m}}\) the set of trees with perfect matchings on 2m vertices. In this paper, we show that for any \({T_1, T_2\in\mathcal{T}_{2m}}\) , if Δ(T 1) > Δ(T 2) and \({\Delta(T_1)\geq \lceil\frac{m}{2}\rceil+2}\) , then μ(T 1) > μ(T 2). By using this result, the first 20th largest trees in \({\mathcal{T}_{2m}}\) according to their Laplacian spectral radius are ordered. We also characterize the tree which alone minimizes (resp., maximizes) the Laplacian spectral radius among all the trees in \({\mathcal{T}_{2m}}\) with an arbitrary fixed maximum degree c (resp., when \({c \geq \lceil\frac{m}{2}\rceil + 1}\)).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, People’s Republic of China

    Xiaodan Chen & Jianguo Qian

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  1. Xiaodan Chen
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  2. Jianguo Qian
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Correspondence to Jianguo Qian.

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Research supported by NSFC(10831001).

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Chen, X., Qian, J. Laplacian Spectral Radius and Maximum Degree of Trees with Perfect Matchings. Graphs and Combinatorics 29, 1249–1257 (2013). https://doi.org/10.1007/s00373-012-1205-8

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  • DOI: https://doi.org/10.1007/s00373-012-1205-8

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