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The Maximum Spectral Radius of Graphs without Spanning Linear Forests

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Abstract

Given a family \(\mathcal {F}\) of graphs, a graph G is called \(\mathcal {F}\)-free if G contains none of \(\mathcal {F}\) as its subgraph. The following problem is one of the most concerned problems in spectral extremal graph theory: what is the maximum spectral radius of an n-vertex \(\mathcal {F}\)-free graph? If each connected component of a graph is either a path (star) or an isolated vertex, then we call it a linear (star) forest. Denote by \(\mathcal {L}_{n,k}\) and \(\mathcal {S}_{n,k}\) the family of all n-vertex linear forests and star forests with k edges, respectively. In this paper, we obtain the maximum spectral radius of an n-vertex \(\mathcal {L}_{n,k}\)-free graph and characterize the extremal graphs based on Kelmans transformation. Also, we obtain the maximum spectral radius of an n-vertex \(\mathcal {S}_{n,k}\)-free graph and characterize the unique extremal graph.

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Acknowledgements

The authors would like to thank three anonymous referees for providing valuable comments and suggestions which improved the presentation of this paper.

Funding

Supported by the National Natural Science Foundation of China (No. 12271439) and China Scholarship Council (No. 202206290003).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Lin-Peng Zhang & Ligong Wang

  2. Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University, Xi’an, 710129, Shaanxi, People’s Republic of China

    Lin-Peng Zhang & Ligong Wang

  3. International Joint Research Center on Operations Research, Optimization and Artificial Intelligence, Xi’an, 710129, Shaanxi, People’s Republic of China

    Lin-Peng Zhang & Ligong Wang

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  1. Lin-Peng Zhang
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  2. Ligong Wang
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Correspondence to Ligong Wang.

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Zhang, LP., Wang, L. The Maximum Spectral Radius of Graphs without Spanning Linear Forests. Graphs and Combinatorics 39, 9 (2023). https://doi.org/10.1007/s00373-022-02608-6

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  • DOI: https://doi.org/10.1007/s00373-022-02608-6

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