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Chromatic number and signless Laplacian spectral radius of graphs. (English) Zbl 1513.05161

Summary: For any simple graph \(G\), the signless Laplacian matrix of \(G\) is defined as \(D(G)+A(G)\), where \(D(G)\) and \(A(G)\) are the diagonal matrix of vertex degrees and the adjacency matrix of \(G\), respectively. Let \(q(G)\) be the signless Laplacian spectral radius of \(G\) (the largest eigenvalue of the signless Laplacian matrix of \(G\)). In this paper we find some relations between the chromatic number and the signless Laplacian spectral radius of graphs. In particular, we characterize all graphs \(G\) of order \(n\) with odd chromatic number \(\chi\) such that \(q(G) = 2n\left(1-\frac{1}{\chi}\right)\). Finally we show that if \(G\) is a graph of order \(n\) and with chromatic number \(\chi\), then under certain conditions, \(q(G) < 2n\left(1-\frac{1}{\chi}\right)-\frac{2}{n}\). This result improves some previous similar results.

MSC:

05C15 Coloring of graphs and hypergraphs
05C31 Graph polynomials
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A18 Eigenvalues, singular values, and eigenvectors
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