On a lower bound for the Laplacian eigenvalues of a graph
If μm and dm denote, respectively, the m-th largest Laplacian eigenvalue and the m-th largest vertex degree of a graph, then μm⩾dm−m+2. This inequality was conjectured by Guo (Linear Multilinear Algebra 55:93–102, 2007) and proved by Brouwer and Haemers (Linear Algebra Appl 429:2131–2135, 2008). Bro...
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sg-ntu-dr.10356-1449962023-02-28T19:53:45Z On a lower bound for the Laplacian eigenvalues of a graph Greaves, Gary Royden Watson Munemasa, Akihiro Peng, Anni School of Physical and Mathematical Sciences Science::Mathematics Laplacian Eigenvalues Degree Sequence If μm and dm denote, respectively, the m-th largest Laplacian eigenvalue and the m-th largest vertex degree of a graph, then μm⩾dm−m+2. This inequality was conjectured by Guo (Linear Multilinear Algebra 55:93–102, 2007) and proved by Brouwer and Haemers (Linear Algebra Appl 429:2131–2135, 2008). Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying μm=dm−m+2. In particular we give a full classification of graphs with μm=dm−m+2⩽1. Accepted version 2020-12-08T03:59:02Z 2020-12-08T03:59:02Z 2017 Journal Article Greaves, G. R. W., Munemasa, A., & Peng, A. (2017). On a lower bound for the Laplacian eigenvalues of a graph. Graphs and Combinatorics, 33(6), 1509-1519. doi:10.1007/s00373-017-1835-y 0911-0119 https://hdl.handle.net/10356/144996 10.1007/s00373-017-1835-y 6 33 1509 1519 en Graphs and Combinatorics © 2017 Springer Japan. This is a post-peer-review, pre-copyedit version of an article published in Graphs and Combinatorics. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00373-017-1835-y application/pdf |
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Science::Mathematics Laplacian Eigenvalues Degree Sequence Greaves, Gary Royden Watson Munemasa, Akihiro Peng, Anni On a lower bound for the Laplacian eigenvalues of a graph |
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If μm and dm denote, respectively, the m-th largest Laplacian eigenvalue and the m-th largest vertex degree of a graph, then μm⩾dm−m+2. This inequality was conjectured by Guo (Linear Multilinear Algebra 55:93–102, 2007) and proved by Brouwer and Haemers (Linear Algebra Appl 429:2131–2135, 2008). Brouwer and Haemers gave several examples of graphs achieving equality, but a complete characterisation was not given. In this paper we consider the problem of characterising graphs satisfying μm=dm−m+2. In particular we give a full classification of graphs with μm=dm−m+2⩽1. |
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School of Physical and Mathematical Sciences |
| author_facet |
School of Physical and Mathematical Sciences Greaves, Gary Royden Watson Munemasa, Akihiro Peng, Anni |
| format |
Article |
| author |
Greaves, Gary Royden Watson Munemasa, Akihiro Peng, Anni |
| author_sort |
Greaves, Gary Royden Watson |
| title |
On a lower bound for the Laplacian eigenvalues of a graph |
| title_short |
On a lower bound for the Laplacian eigenvalues of a graph |
| title_full |
On a lower bound for the Laplacian eigenvalues of a graph |
| title_fullStr |
On a lower bound for the Laplacian eigenvalues of a graph |
| title_full_unstemmed |
On a lower bound for the Laplacian eigenvalues of a graph |
| title_sort |
on a lower bound for the laplacian eigenvalues of a graph |
| publishDate |
2020 |
| url |
https://hdl.handle.net/10356/144996 |
| _version_ |
1759856010846011392 |