Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen; Guoliang Hao; Dequan Jin; Jingjian Li

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 601-610
  • ISSN: 0011-4642

Abstract

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For a simple graph G on n vertices and an integer k with 1 ≤ k ≤ n , denote by 𝒮 k + ( G ) the sum of k largest signless Laplacian eigenvalues of G . It was conjectured that 𝒮 k + ( G ) ≤ e ( G ) + k + 1 2 , where e ( G ) is the number of edges of G . This conjecture has been proved to be true for all graphs when k ∈ { 1 , 2 , n - 1 , n } , and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all k ). In this note, this conjecture is proved to be true for all graphs when k = n - 2 , and for some new classes of graphs.

How to cite

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Chen, Xiaodan, et al. "Note on a conjecture for the sum of signless Laplacian eigenvalues." Czechoslovak Mathematical Journal 68.3 (2018): 601-610. <http://eudml.org/doc/294439>.

@article{Chen2018,
abstract = {For a simple graph $G$ on $n$ vertices and an integer $k$ with $1\le k\le n$, denote by $\mathcal \{S\}_k^+(G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$. It was conjectured that $\mathcal \{S\}_k^+(G)\le e(G)+\{k+1 \atopwithdelims ()2\}$, where $e(G)$ is the number of edges of $G$. This conjecture has been proved to be true for all graphs when $k\in \lbrace 1,2,n-1,n\rbrace $, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all $k$). In this note, this conjecture is proved to be true for all graphs when $k=n-2$, and for some new classes of graphs.},
author = {Chen, Xiaodan, Hao, Guoliang, Jin, Dequan, Li, Jingjian},
journal = {Czechoslovak Mathematical Journal},
keywords = {sum of signless Laplacian eigenvalues; upper bound; clique number; girth},
language = {eng},
number = {3},
pages = {601-610},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Note on a conjecture for the sum of signless Laplacian eigenvalues},
url = {http://eudml.org/doc/294439},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Chen, Xiaodan
AU - Hao, Guoliang
AU - Jin, Dequan
AU - Li, Jingjian
TI - Note on a conjecture for the sum of signless Laplacian eigenvalues
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 601
EP - 610
AB - For a simple graph $G$ on $n$ vertices and an integer $k$ with $1\le k\le n$, denote by $\mathcal {S}_k^+(G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$. It was conjectured that $\mathcal {S}_k^+(G)\le e(G)+{k+1 \atopwithdelims ()2}$, where $e(G)$ is the number of edges of $G$. This conjecture has been proved to be true for all graphs when $k\in \lbrace 1,2,n-1,n\rbrace $, and for trees, unicyclic graphs, bicyclic graphs and regular graphs (for all $k$). In this note, this conjecture is proved to be true for all graphs when $k=n-2$, and for some new classes of graphs.
LA - eng
KW - sum of signless Laplacian eigenvalues; upper bound; clique number; girth
UR - http://eudml.org/doc/294439
ER -

References

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