×

Upper bounds for the largest singular value of certain digraph matrices. (English) Zbl 1462.05161

Summary: 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.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bollobás, B.; Nikiforov, V., Graphs and Hermitian matrices: eigenvalue interlacing, Discrete Math., 289, 119-127 (2004) · Zbl 1063.05088
[2] Brualdi, R., Spectra of digraphs, Linear Algebra Appl., 432, 2181-2213 (2010) · Zbl 1221.05177
[3] Chat, BA; Ganie, HA; Pirzada, S., Bounds for the skew Laplacian spectral radius of oriented graphs, Carpathian J. Math., 35, 31-40 (2019) · Zbl 1463.05324
[4] Ganie, HA, Bounds for the skew Laplacian (skew adjacency) spectral radius of a digraph, Trans. Comb., 8, 1-12 (2019) · Zbl 1463.05222
[5] Guo, J-M, A new upper bound for the Laplacian spectral radius of graphs, Linear Algebra Appl., 400, 61-66 (2005) · Zbl 1062.05091
[6] Favaron, O.; Mahéo, M.; Saclé, J-F, Some eigenvalue properties in graphs (conjectures of Graffiti—II), Discrete Math., 111, 197-220 (1993) · Zbl 0785.05065
[7] Stanić, Z., Inequalities for Graph Eigenvalues (2015), Cambridge: Cambridge University Press, Cambridge · Zbl 1368.05001
[8] Xu, G-H, Some inequalities on the skew-spectral radii of oriented graphs, J. Inequal. Appl., 2012, 211 (2012) · Zbl 1277.05113
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.