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Spectral Extremal Results with Forbidding Linear Forests

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Abstract

The Turán type extremal problems ask to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, their spectral counterparts ask to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without a linear forest as a subgraph and all the extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without \(k\cdot P_3\) as a subgraph are obtained and all the extremal graphs are also determined. Moreover, some relations between Turán type extremal problems and their spectral counterparts are discussed.

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Acknowledgements

The authors would like to thank the anonymous referee for many helpful and constructive suggestions to an earlier version of this paper, which results in an great improvement.

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Authors and Affiliations

  1. School of Mathematical Sciences, MOE-LSC, SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Ming-Zhu Chen, A-Ming Liu & Xiao-Dong Zhang

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  1. Ming-Zhu Chen
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  2. A-Ming Liu
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  3. Xiao-Dong Zhang
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Corresponding author

Correspondence to Xiao-Dong Zhang.

Additional information

This work is supported by the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (no. 11561141001)), the National Natural Science Foundation of China (no. 11531001), and the Weng Hongwu Research Foundation of Peking University (no. WHW201803).

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Chen, MZ., Liu, AM. & Zhang, XD. Spectral Extremal Results with Forbidding Linear Forests. Graphs and Combinatorics 35, 335–351 (2019). https://doi.org/10.1007/s00373-018-1996-3

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  • DOI: https://doi.org/10.1007/s00373-018-1996-3

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