Remarks on spectral radius and Laplacian eigenvalues of a graph

@article{Zhou2005,
abstract = {Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1, d_2, \dots , d_n)$, where $d_1 \ge d_2 \ge \dots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho (G)$ and $\mu (G)$, respectively. We determine the graphs with \[ \rho (G) = \frac\{d\_n - 1\}\{2\} + \sqrt\{2m - nd\_n + \frac\{(d\_n +1)^2\}\{4\}\} \] and the graphs with $d_n\ge 1$ and \[ \mu (G) = d\_n + \frac\{1\}\{2\} + \sqrt\{\sum \_\{i=1\}^n d\_i (d\_i-d\_n) + \Bigl (d\_n - \frac\{1\}\{2\} \Bigr )^2\}. \] We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.},
author = {Zhou, Bo, Cho, Han Hyuk},
journal = {Czechoslovak Mathematical Journal},
keywords = {spectral radius; Laplacian eigenvalue; strongly regular graph; spectral radius; Laplacian eigenvalue; strongly regular graph},
language = {eng},
number = {3},
pages = {781-790},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on spectral radius and Laplacian eigenvalues of a graph},
url = {http://eudml.org/doc/30987},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Zhou, Bo
AU - Cho, Han Hyuk
TI - Remarks on spectral radius and Laplacian eigenvalues of a graph
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 3
SP - 781
EP - 790
AB - Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1, d_2, \dots , d_n)$, where $d_1 \ge d_2 \ge \dots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho (G)$ and $\mu (G)$, respectively. We determine the graphs with \[ \rho (G) = \frac{d_n - 1}{2} + \sqrt{2m - nd_n + \frac{(d_n +1)^2}{4}} \] and the graphs with $d_n\ge 1$ and \[ \mu (G) = d_n + \frac{1}{2} + \sqrt{\sum _{i=1}^n d_i (d_i-d_n) + \Bigl (d_n - \frac{1}{2} \Bigr )^2}. \] We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.
LA - eng
KW - spectral radius; Laplacian eigenvalue; strongly regular graph; spectral radius; Laplacian eigenvalue; strongly regular graph
UR - http://eudml.org/doc/30987
ER -