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On the skew Laplacian spectral radius of a digraph. (English) Zbl 1493.05187

Summary: Let \(\mathbf{G}\) be an orientation of a simple graph \(G\) with \(n\) vertices and \(m\) edges. The skew Laplacian matrix \(\mathrm{SL}(\mathbf{G})\) of the digraph \(\mathbf{G}\) is defined as \(\mathrm{SL}(\mathbf{G})= \widetilde{D}(\mathbf{G})-iS(\mathbf{G})\), where \(i=\sqrt{ - 1}\) is the imaginary unit, \( \widetilde{D}(\mathbf{G})\) is the diagonal matrix with oriented degrees \(\alpha_i= d_i^+- d_i^-\) as diagonal entries and \(S(\mathbf{G})\) is the skew matrix of the digraph \(\mathbf{G}\). The largest eigenvalue of the matrix \(\mathrm{SL}(\mathbf{G})\) is called skew Laplacian spectral radius of the digraph \(\mathbf{G}\). In this paper, we study the skew Laplacian spectral radius of the digraph \(\mathbf{G}\). We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph \(\mathbf{G}\), in terms of different structural parameters of the digraph and the underlying graph. We characterize the extremal digraphs attaining these bounds in some cases. Further, we end the paper with some problems for the future research in this direction.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C12 Distance in graphs
05C20 Directed graphs (digraphs), tournaments
15A18 Eigenvalues, singular values, and eigenvectors
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