In this paper, we consider digraphs with possible loops and the particular case of oriented graphs, i.e. loopless digraphs with at most one oriented edge between every pair of vertices. We provide an upper bound for the largest singular value of the skew Laplacian matrix of an oriented graph, the largest singular value of the skew adjacency matrix of an oriented graph and the largest singular value of the adjacency matrix of a digraph. These bounds are expressed in terms of certain parameters related to vertex degrees. We also consider some bounds for the sums of squares of singular values. As an application, for the skew (Laplacian) adjacency matrix of an oriented graph and the adjacency matrix of a digraph, we derive some upper bounds for the spectral radius and the sums of squares of moduli of eigenvalues.
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Bollobás, B., Nikiforov, V.: Graphs and Hermitian matrices: eigenvalue interlacing. Discrete Math. 289, 119–127 (2004)
Brualdi, R.: Spectra of digraphs. Linear Algebra Appl. 432, 2181–2213 (2010)
Chat, B.A., Ganie, H.A., Pirzada, S.: Bounds for the skew Laplacian spectral radius of oriented graphs. Carpathian J. Math. 35, 31–40 (2019)
Ganie, H.A.: Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph. Trans. Comb. 8, 1–12 (2019)
Guo, J.-M.: A new upper bound for the Laplacian spectral radius of graphs. Linear Algebra Appl. 400, 61–66 (2005)
Favaron, O., Mahéo, M., Saclé, J.-F.: Some eigenvalue properties in graphs (conjectures of Graffiti—II). Discrete Math. 111, 197–220 (1993)
Stanić, Z.: Inequalities for Graph Eigenvalues. Cambridge University Press, Cambridge (2015)
Xu, G.-H.: Some inequalities on the skew-spectral radii of oriented graphs. J. Inequal. Appl. 2012, 211 (2012)
Research is partially supported by Serbian Ministry of Education, Science and Technological Development via University of Belgrade.
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Communicated by Sanming Zhou.
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Stanić, Z. Upper Bounds for the Largest Singular Value of Certain Digraph Matrices. Bull. Malays. Math. Sci. Soc. (2020). https://doi.org/10.1007/s40840-020-00970-3
- Oriented graph
- (skew) adjacency matrix
- Skew Laplacian matrix
- Singular value
- Upper bound
Mathematics Subject Classification