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On conjectures involving second largest signless Laplacian eigenvalue of graphs. (English) Zbl 1195.05040

Summary: Let \(G=(V,E)\) be a simple graph. Denote by \(D(G)\) the diagonal matrix of its vertex degrees and by \(A(G)\) its adjacency matrix. Then the Laplacian matrix of \(G\) is \(L(G)=D(G)-A(G)\) and the signless Laplacian matrix of \(G\) is \(Q(G)=D(G)+A(G)\). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of \(G\). In [5], Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of \(G\) (see also [1]). Here we prove five conjectures.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C35 Extremal problems in graph theory
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References:

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