Bounds for the (Laplacian) spectral radius of graphs with parameter α

Gui-Xian Tian; Ting-Zhu Huang

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 567-580
  • ISSN: 0011-4642

Abstract

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Let G be a simple connected graph of order n with degree sequence ( d 1 , d 2 , ... , d n ) . Denote ( α t ) i = ∑ j : i ∼ j d j α , ( α m ) i = ( α t ) i / d i α and ( α N ) i = ∑ j : i ∼ j ( α t ) j , where α is a real number. Denote by λ 1 ( G ) and μ 1 ( G ) the spectral radius of the adjacency matrix and the Laplacian matrix of G , respectively. In this paper, we present some upper and lower bounds of λ 1 ( G ) and μ 1 ( G ) in terms of ( α t ) i , ( α m ) i and ( α N ) i . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

How to cite

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Tian, Gui-Xian, and Huang, Ting-Zhu. "Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $." Czechoslovak Mathematical Journal 62.2 (2012): 567-580. <http://eudml.org/doc/246489>.

@article{Tian2012,
abstract = {Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _\{j\colon i \sim j\} \{d_j^\alpha \}$, $(^\alpha m)_i = \{(^\alpha t)_i \}/\{d_i^\alpha \}$ and $(^\alpha N)_i = \sum \nolimits _\{j\colon i \sim j\} \{(^\alpha t)_j \}$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.},
author = {Tian, Gui-Xian, Huang, Ting-Zhu},
journal = {Czechoslovak Mathematical Journal},
keywords = {graph; adjacency matrix; Laplacian matrix; spectral radius; bound; adjacency matrix; Laplacian matrix; spectral radius},
language = {eng},
number = {2},
pages = {567-580},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $},
url = {http://eudml.org/doc/246489},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Tian, Gui-Xian
AU - Huang, Ting-Zhu
TI - Bounds for the (Laplacian) spectral radius of graphs with parameter $\alpha $
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 567
EP - 580
AB - Let $G$ be a simple connected graph of order $n$ with degree sequence $(d_1,d_2,\ldots ,d_n)$. Denote $(^\alpha t)_i = \sum \nolimits _{j\colon i \sim j} {d_j^\alpha }$, $(^\alpha m)_i = {(^\alpha t)_i }/{d_i^\alpha }$ and $(^\alpha N)_i = \sum \nolimits _{j\colon i \sim j} {(^\alpha t)_j }$, where $\alpha $ is a real number. Denote by $\lambda _1(G)$ and $\mu _1(G)$ the spectral radius of the adjacency matrix and the Laplacian matrix of $G$, respectively. In this paper, we present some upper and lower bounds of $\lambda _1(G)$ and $\mu _1(G)$ in terms of $(^\alpha t)_i $, $(^\alpha m)_i $ and $(^\alpha N)_i $. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.
LA - eng
KW - graph; adjacency matrix; Laplacian matrix; spectral radius; bound; adjacency matrix; Laplacian matrix; spectral radius
UR - http://eudml.org/doc/246489
ER -

References

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