Geodesic
Note on a conjecture for the sum of signless Laplacian eigenvalues
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 601-610 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $\mathcal {S}_k^+(G)\leq e(G)+{k+1 \choose 2}$, where $e(G)$ is the number of edges of $G$. This conjecture has been proved to be true for all graphs when $k\in \{1,2,n-1,n\}$, 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.
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 $\mathcal {S}_k^+(G)\leq e(G)+{k+1 \choose 2}$, where $e(G)$ is the number of edges of $G$. This conjecture has been proved to be true for all graphs when $k\in \{1,2,n-1,n\}$, 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.
DOI : 10.21136/CMJ.2018.0548-16
Classification : 05C50, 15A18
Keywords: sum of signless Laplacian eigenvalues; upper bound; clique number; girth 
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Chen, Xiaodan; Hao, Guoliang; Jin, Dequan; Li, Jingjian. Note on a conjecture for the sum of signless Laplacian eigenvalues. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 601-610. doi: 10.21136/CMJ.2018.0548-16

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