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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  1. Ashraf, F., Omidi, G. R., Tayfeh-Rezaie, B., 10.1016/j.laa.2013.01.023, Linear Algebra Appl. 438 (2013), 4539-4546. (2013) Zbl1282.05087MR3034549DOI10.1016/j.laa.2013.01.023
  2. Bai, H., 10.1090/S0002-9947-2011-05393-6, Trans. Am. Math. Soc. 363 (2011), 4463-4474. (2011) Zbl1258.05066MR2792996DOI10.1090/S0002-9947-2011-05393-6
  3. Brouwer, A. E., Haemers, W. H., 10.1007/978-1-4614-1939-6, Universitext, Springer, Berlin (2012). (2012) Zbl1231.05001MR2882891DOI10.1007/978-1-4614-1939-6
  4. Cvetković, D., Rowlinson, P., Simić, S., 10.1017/CBO9780511801518, London Mathematical Society Student Texts 75, Cambridge University Press, Cambridge (2010). (2010) Zbl1211.05002MR2571608DOI10.1017/CBO9780511801518
  5. Du, Z., Zhou, B., 10.1016/j.laa.2012.01.007, Linear Algebra Appl. 436 (2012), 3672-3683. (2012) Zbl1241.05074MR2900744DOI10.1016/j.laa.2012.01.007
  6. Feng, L., Yu, G., 10.2298/PIM0999035F, Publ. Inst. Math., Nouv. Sér. 85 (2009), 35-38. (2009) Zbl1265.05365MR2536687DOI10.2298/PIM0999035F
  7. Fritscher, E., Hoppen, C., Rocha, I., Trevisan, V., 10.1016/j.laa.2011.01.036, Linear Algebra Appl. 435 (2011), 371-399. (2011) Zbl1226.05154MR2782788DOI10.1016/j.laa.2011.01.036
  8. Ganie, H. A., Alghamdi, A. M., Pirzada, S., 10.1016/j.laa.2016.03.034, Linear Algebra Appl. 501 (2016), 376-389. (2016) Zbl1334.05080MR3485073DOI10.1016/j.laa.2016.03.034
  9. Grone, R., Merris, R., 10.1137/S0895480191222653, SIAM J. Discrete Math. 7 (1994), 221-229. (1994) Zbl0795.05092MR1271994DOI10.1137/S0895480191222653
  10. Haemers, W. H., Mohammadian, A., Tayfeh-Rezaie, B., 10.1016/j.laa.2009.03.038, Linear Algebra Appl. 432 (2010), 2214-2221. (2010) Zbl1218.05094MR2599854DOI10.1016/j.laa.2009.03.038
  11. Rocha, I., Trevisan, V., 10.1016/j.dam.2014.01.023, Discrete Appl. Math. 170 (2014), 95-103. (2014) Zbl1288.05167MR3176708DOI10.1016/j.dam.2014.01.023
  12. Wang, S., Huang, Y., Liu, B., 10.1016/j.mcm.2011.12.047, Math. Comput. Modelling 56 (2012), 60-68. (2012) Zbl1255.05118MR2935294DOI10.1016/j.mcm.2011.12.047
  13. Yang, J., You, L., 10.1016/j.laa.2013.12.032, Linear Algebra Appl. 446 (2014), 115-132. (2014) Zbl1292.05182MR3163132DOI10.1016/j.laa.2013.12.032

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