×

Hypoenergetic and nonhypoenergetic digraphs. (English) Zbl 1462.05209

Summary: The energy of a graph \(G, \mathcal{E}(G)\), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by V. Nikiforov [ibid. 506, 82–138 (2016; Zbl 1344.05089)] to arbitrary complex matrices. Recall that the trace norm of a digraph \(D\) is defined as, \( \mathcal{N}(D) = \sum_{i = 1}^n \sigma_i\), where \(\sigma_1 \geq \cdots \geq \sigma_n\) are the singular values of the adjacency matrix of \(D\). In this paper we would like to present some lower and upper bounds for \(\mathcal{N}(D)\). For any digraph \(D\) it is proved that \(\mathcal{N}(D) \geq \operatorname{rank} ( \operatorname{D} )\) and the equality holds if and only if \(D\) is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on \(\sigma_1\) and \(\mathcal{N}(D)\) in terms of the size of digraph \(D\).

MSC:

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

Citations:

Zbl 1344.05089
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agudelo, N.; de la Peña, J. A.; Rada, J., Extremal values of the trace norm over oriented trees, Linear Algebra Appl., 505, 261-268 (2016) · Zbl 1338.05153
[2] Agudelo, N.; Rada, J., Lower bounds of Nikiforov’s energy over digraphs, Linear Algebra Appl., 494, 156-164 (2016) · Zbl 1331.05133
[3] Akbari, S.; Ghahremani, M.; Gutman, I.; Koorepazan-Moftakhar, F., Orderenergetic graphs, MATCH Commun. Math. Comput. Chem., 84, 325-334 (2020)
[4] Cvetković, D.; Rowlinson, P.; Simić, S., An Introduction to the Theory of Graph Spectra, London Mathematical Society Student Texts, vol. 75 (2010), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1211.05002
[5] Gutman, I., The energy of a graph, Ber. Math. Statist. Sekt. Forschungsz. Graz., 103, 1-22 (1978)
[6] Gutman, I.; Radenkovic, S., Hypoenergetic molecular graphs, Indian J. Chem., 46A, 1733-1736 (2007)
[7] Horn, R.; Johnson, C., Matrix Analysis (1989), Cambridge University Press: Cambridge University Press Cambridge
[8] Mirsky, L., The spread of a matrix, Mathematika, 3, 127-130 (1956) · Zbl 0073.00903
[9] Monsalve, J.; Rada, J., Oriented bipartite graphs with minimal trace norm, Linear Multilinear Algebra, 67, 6, 1121-1131 (2019) · Zbl 1411.05172
[10] Nikiforov, V., The energy of graphs and matrices, J. Math. Anal. Appl., 326, 1472-1475 (2007) · Zbl 1113.15016
[11] Nikiforov, V., Beyond graph energy: norms of graphs and matrices, Linear Algebra Appl., 506, 82-138 (2016) · Zbl 1344.05089
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.