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