On graphs with the largest Laplacian index

Liu, Bo Lian, Chen, Zhibo, and Liu, Muhuo. "On graphs with the largest Laplacian index." Czechoslovak Mathematical Journal 58.4 (2008): 949-960. <http://eudml.org/doc/37879>.

@article{Liu2008,
abstract = {Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \ge 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.},
author = {Liu, Bo Lian, Chen, Zhibo, Liu, Muhuo},
journal = {Czechoslovak Mathematical Journal},
keywords = {eigenvalue; Laplacian index; algebraic connectivity; semi-regular graph; regular graph; Hamiltonian graph; planar graph; eigenvalue; Laplacian index; algebraic connectivity; semi-regular graph; regular graph; Hamiltonian graph; planar graph},
language = {eng},
number = {4},
pages = {949-960},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On graphs with the largest Laplacian index},
url = {http://eudml.org/doc/37879},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Liu, Bo Lian
AU - Chen, Zhibo
AU - Liu, Muhuo
TI - On graphs with the largest Laplacian index
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 949
EP - 960
AB - Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \ge 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.
LA - eng
KW - eigenvalue; Laplacian index; algebraic connectivity; semi-regular graph; regular graph; Hamiltonian graph; planar graph; eigenvalue; Laplacian index; algebraic connectivity; semi-regular graph; regular graph; Hamiltonian graph; planar graph
UR - http://eudml.org/doc/37879
ER -