Abstract
Let G be a simple graph and let \(\beta (G)\) be the matching number of G. It is well-known that \({{\,\mathrm{reg}\,}}I(G) \leqslant \beta (G)+1\). In this paper we show that \({{\,\mathrm{reg}\,}}I(G) = \beta (G)+1\) if and only if every connected component of G is either a pentagon or a Cameron–Walker graph.
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Acknowledgements
This work is partially supported by NAFOSTED (Vietnam) under the Grant No. 101.04 - 2018.307. Part of this work was done while I was at the Vietnam Institute of Advanced Studies in Mathematics (VIASM) in Hanoi, Vietnam. I would like to thank VIASM for its hospitality. I would also like to thank an anonymous referee for many helpful comments.
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Trung, T.N. Regularity, matchings and Cameron–Walker graphs. Collect. Math. 71, 83–91 (2020). https://doi.org/10.1007/s13348-019-00250-9
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DOI: https://doi.org/10.1007/s13348-019-00250-9