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On skew Laplacian spectrum and energy of digraphs. (English) Zbl 1458.05142

Summary: We consider the skew Laplacian matrix of a digraph \(\vec{G}\) obtained by giving an arbitrary direction to the edges of a graph \(G\) having \(n\) vertices and \(m\) edges. With \(\nu_1,\nu_2,\dots,\nu_n\) to be the skew Laplacian eigenvalues of \(\vec{G}\), the skew Laplacian energy \(\text{SLE}(\vec{G})\) of \(\vec{G}\) is defined as \(\text{SLE}(\vec{G})=\sum_{i=1}^n|\nu_i|\). In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy \(\text{SLE}(\vec{G})\) in terms of various parameters associated with the digraph \(\vec{G}\) and the underlying graph \(G\) and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C20 Directed graphs (digraphs), tournaments
15B36 Matrices of integers
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References:

[1] Adiga, C., Balakrishnan, R. and So, W., The skew energy of a digraph, Linear Algebra Appl.432 (2010) 1825-1835. · Zbl 1217.05131
[2] Cvetkovic, D., Doob, M. and Sachs, H., Spectra of Graphs-Theory and Application (Academic Press, New York, 1980). · Zbl 0458.05042
[3] Cai, Q., Li, X. and Song, J., New skew Laplacian energy of simple digraphs, Trans. Combin.2(1) (2013) 27-37. · Zbl 1316.05085
[4] Ganie, H. A. and Pirzada, S., On the bounds for signless Laplacian energy of a graph, Discr. Appl. Math.228 (2017) 3-13. · Zbl 1365.05172
[5] Ganie, H. A., Alghamdi, A. M. and Pirzada, S., On the sum of the Laplacian eigenvalues of a graph and Brouwer’s conjecture, Linear Algebra Appl.501 (2016) 376-389. · Zbl 1334.05080
[6] Ganie, H. A., Chat, B. A. and Pirzada, S., Signless Laplacian energy of a graph and energy of a line graph, Linear Algebra Appl.544 (2018) 306-324. · Zbl 1388.05114
[7] H. A. Ganie, B. A. Chat and S. Pirzada, On real or integral skew Laplacian spectrum of digraphs, Preprint.
[8] Ganie, H. A., Chat, B. A. and Pirzada, S., Skew Laplacian energy of digraphs, Kragujevac J. Maths.43(1) (2019) 87-98.
[9] Horn, R. and Johnson, C., Matrix Analysis (Cambridge University Press, 1985). · Zbl 0576.15001
[10] Horn, R. and Johnson, C., Topics in Matrix Analysis (Cambridge University Press, 1991). · Zbl 0729.15001
[11] X. Li and H. Lian, A survey on the skew energy of oriented graphs, arXiv:1304.5707v6 [math.CO] 18 May 2015.
[12] Pirzada, S. and Ganie, H. A., On the Laplacian eigenvalues of a graph and Laplacian energy, Linear Algebra Appl.486 (2015) 454-468. · Zbl 1327.05157
[13] Pirzada, S., An Introduction to Graph Theory (Universities Press, Orient BlackSwan, Hyderabad, India, 2012).
[14] Ramane, H. S., Nandeesh, K. C., Gutman, I. and Li, X., Skew equienergetic digraphs, Trans. Combin.5(1) (2016) 15-23. · Zbl 1463.05228
[15] Xu, G., Some inequalities on the skew-spectral radii of oriented graphs, J. Inequal. Appl.2020 (2012) 211. · Zbl 1277.05113
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