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The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs

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Abstract

In this paper, we study the relationship between spectral radius and maximum average degree of graphs. By using this relationship and the previous technique of Li and Ning in (J Graph Theory 103:486–492, 2023), we prove that, for any given positive number \(\varepsilon <\frac{1}{3}\), if n is a sufficiently large integer, then any graph G of order n with \(\rho (G)>\sqrt{\left\lfloor \frac{n^{2}}{4}\right\rfloor }\) contains a cycle of length t for all integers \(t\in [3,(\frac{1}{3}-\varepsilon )n]\), where \(\rho (G)\) is the spectral radius of G. This improves the result of Li and Ning (2023).

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Acknowledgements

The author would like to thank the editors and the referees for their valuable comments.

Funding

This work is supported by NSFC (No. 12301448) and NSF of Shandong Province (No. ZR2022QA045).

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  1. School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255000, Shandong, P.R. China

    Wenqian Zhang

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  1. Wenqian Zhang
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Correspondence to Wenqian Zhang.

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Zhang, W. The Spectral Radius, Maximum Average Degree and Cycles of Consecutive Lengths of Graphs. Graphs and Combinatorics 40, 32 (2024). https://doi.org/10.1007/s00373-024-02761-0

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  • DOI: https://doi.org/10.1007/s00373-024-02761-0

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