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Spectral Radius and Fractional Perfect Matchings in Graphs

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Abstract

For an n-vertex graph G, a fractional matching of G is a function f giving each edge a real number in [0, 1] such that \(\sum _{e\in \Gamma (v)}f(e)\le 1\) for each vertex \(v\in V(G)\), where \(\Gamma (v)\) is the set of edges incident to v. A fractional perfect matching is a fractional matching f with \(\sum _{e\in E(G)}f(e)=\frac{n}{2}\). In this paper, we establish tight lower bounds on the size and the spectral radius of G to guarantee that G has a fractional perfect matching, respectively. In addition, we investigate the relationship between fractional perfect matching and \(\mathcal {P}_{\geqslant 2}\)-factor, and give some sufficient conditions for a graph to have a \(\mathcal {P}_{\geqslant 2}\)-factor.

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Acknowledgements

The authors would like to express their sincere gratitude to all the referees for their careful reading and insightful suggestions.

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Authors and Affiliations

  1. 63763 Army of PLA, Lingshui, 572400, China

    Yingui Pan

  2. College of Sciences, National University of Defense Technology, Changsha, 410073, China

    Chang Liu

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  1. Yingui Pan
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  2. Chang Liu
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Correspondence to Yingui Pan.

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Pan, Y., Liu, C. Spectral Radius and Fractional Perfect Matchings in Graphs. Graphs and Combinatorics 39, 52 (2023). https://doi.org/10.1007/s00373-023-02652-w

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  • DOI: https://doi.org/10.1007/s00373-023-02652-w

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