On the multiplicity of Laplacian eigenvalues for unicyclic graphs

Wen, Fei, and Huang, Qiongxiang. "On the multiplicity of Laplacian eigenvalues for unicyclic graphs." Czechoslovak Mathematical Journal 72.2 (2022): 371-390. <http://eudml.org/doc/298311>.

@article{Wen2022,
abstract = {Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_\{G\}(\mu )$, the multiplicity of a Laplacian eigenvalue $\mu $ of $G$. As a straightforward result, $m_\{U\}(1)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_\{G\}(1)$ is nondecreasing. As applications, we get the distribution of $m_\{U\}(1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_\{U\}(1)\in \lbrace n-5,n-3\rbrace $, the corresponding graphs $U$ are completely determined.},
author = {Wen, Fei, Huang, Qiongxiang},
journal = {Czechoslovak Mathematical Journal},
keywords = {unicyclic graph; Laplacian eigenvalue; multiplicity; bound},
language = {eng},
number = {2},
pages = {371-390},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the multiplicity of Laplacian eigenvalues for unicyclic graphs},
url = {http://eudml.org/doc/298311},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Wen, Fei
AU - Huang, Qiongxiang
TI - On the multiplicity of Laplacian eigenvalues for unicyclic graphs
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 2
SP - 371
EP - 390
AB - Let $G$ be a connected graph of order $n$ and $U$ a unicyclic graph with the same order. We firstly give a sharp bound for $m_{G}(\mu )$, the multiplicity of a Laplacian eigenvalue $\mu $ of $G$. As a straightforward result, $m_{U}(1)\le n-2$. We then provide two graph operations (i.e., grafting and shifting) on graph $G$ for which the value of $m_{G}(1)$ is nondecreasing. As applications, we get the distribution of $m_{U}(1)$ for unicyclic graphs on $n$ vertices. Moreover, for the two largest possible values of $m_{U}(1)\in \lbrace n-5,n-3\rbrace $, the corresponding graphs $U$ are completely determined.
LA - eng
KW - unicyclic graph; Laplacian eigenvalue; multiplicity; bound
UR - http://eudml.org/doc/298311
ER -