Note on a conjecture for the sum of signless Laplacian eigenvalues |
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Authors: | Xiaodan Chen Guoliang Hao Dequan Jin Jingjian Li |
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Institution: | 1.College of Mathematics and Information Science,Guangxi University,Nanning, Guangxi,P.R. China;2.College of Science,East China University of Technology,Nanchang, Jiangxi,P.R. China |
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Abstract: | For a simple graph G on n vertices and an integer k with 1 ? k ? n, denote by \(\mathcal{S}^+_k\) (G) the sum of k largest signless Laplacian eigenvalues of G. It was conjectured that \(\mathcal{S}^+_k(G)\leqslant{e}(G)+(^{k+1}_{2})\) (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. |
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