Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 601-610, 2018

Note on a conjecture for the sum of signless Laplacian eigenvalues

Xiaodan Chen, Guoliang Hao, Dequan Jin, Jingjian Li

Received October 18, 2016. Published online June 7, 2018.

Abstract: For a simple graph $G$ on $n$ vertices and an integer $k$ with $1\leq k\leq n$, denote by $\mathcal{S}_k^+(G)$ the sum of $k$ largest signless Laplacian eigenvalues of $G$. It was conjectured that $\mathcal{S}_k^+(G)\leq e(G)+{k+1 \choose2}$, where $e(G)$ is the number of edges of $G$. This conjecture has been proved to be true for all graphs when $k\in\{1,2,n-1,n\}$, 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.

Keywords: sum of signless Laplacian eigenvalues; upper bound; clique number; girth

Classification MSC: 05C50, 15A18

DOI: 10.21136/CMJ.2018.0548-16

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Xiaodan Chen, College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China, e-mail: x.d.chen@live.cn; Guoliang Hao (corresponding author), College of Science, East China University of Technology, No. 418, Guanglan Road, Nanchang 330013, Jiangxi, P. R. China, e-mail: guoliang-hao@163.com; Dequan Jin, Jingjian Li, College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China, e-mail: dqjin@yahoo.cn, lijjhx@163.com

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