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Bounds for the skew Laplacian energy of weighted digraphs. (English) Zbl 1488.05295

Summary: Let \(\mathbb{D}\) be a simple connected digraph with \(n\) vertices and \(m\) arcs and let \(W(\mathbb{D})=(\mathbb{D},\omega)\) be the weighted digraph corresponding to \(\mathbb{D}\), where the weights are taken from the set of non-zero real numbers. In this paper, we define the skew Laplacian matrix \(SL(W(\mathbb{D}))\) and skew Laplacian energy \(SLE(W(\mathbb{D}))\) of a weighted digraph \(W(\mathbb{D})\), which is defined as the sum of the absolute values of the skew Laplacian eigenvalues, that is, \(SLE(W(\mathbb{D}))=\sum_{i=1}^n|\rho_i|\), where \(\rho_1,\rho_2, \dots ,\rho_n\) are the skew Laplacian eigenvalues of \(W(\mathbb{D})\). We show the existence of the real skew Laplacian eigenvalues of a weighted digraph when the weighted digraph has an independent set such that all the vertices in the independent set have the same out-neighbors and in-neighbors. We obtain a Koolen type upper bound for \(SLE(W(\mathbb{D}))\). Further, for a connected weighted digraph \(W(\mathbb{D})\), we obtain bounds for \(SLE(W(\mathbb{D}))\), in terms of different digraph parameters associated with the digraph structure \(\mathbb{D} \). We characterize the extremal weighted digraphs attaining these bounds.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C20 Directed graphs (digraphs), tournaments
05C22 Signed and weighted graphs
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