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Abstract
Let 𝒰(n, d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n, d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].
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Acknowledgements
Many thanks to the anonymous referees and editor Jiayu Shao for their useful comments and suggestions, which have considerably improved the presentation of the article. M. Zhai was supported by the National Natural Science Foundation of China (No. 10971248) and the Foundation for the Excellent Young Talents of Anhui Province (No. 2010SQRL136ZD) and J. Shu was supported by the National Natural Science Foundation of China (No. 10671074), Open Research Funding Program of LGISEM and Shanghai Leading Academic Discipline Project (B407).