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Matching Extendability and Connectivity of Regular Graphs from Eigenvalues

  • Wenqian ZhangEmail author
Original Paper
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Abstract

Let G be a graph on even number of vertices. A perfect matching of G is a set of independent edges which cover each vertex of G. For an integer \(t\ge 1\), G is t-extendable if G has a perfect matching, and for any t independent edges, G has a perfect matching which contains these given t edges. The graph G is bi-critical if for any two vertices u and v, the graph \(G-\left\{ u,v\right\} \) contains a perfect matching. Let G be a connected k-regular graph. In this paper, we obtain two sharp sufficient eigenvalue conditions for a connected k-regular graph to be 1-extendable and bi-critical, respectively. Our results are in term of the second largest eigenvalue of the adjacency matrix of G. We also give examples that show that there is no good spectral characterization (in term of the second largest eigenvalue of the adjacency matrix) for 2-extendability of regular graphs in general. Also, for any integer \(1\le \ell <\frac{k+2}{2}\), we obtain an eigenvalue condition for G to be \((\ell +1)\)-connected.

Keywords

Regular graph Eigenvalue 1-Extendable Bicritical Connectivity 

Mathematics Subject Classification

05C40 05C50 05C70 
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Notes

Acknowledgements

The author would like to thank the editors and the anonymous referees for their helpful comments on improving the representation of the paper.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingChina

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