The DOI matches its Crossref record; 9 of the 9 references it cites were found in a registry or an index.
The energy of graphs and matrices
Open accessAbstract
Abstract Given a complex m × n matrix A , we index its singular values as σ 1 ( A ) ⩾ σ 2 ( A ) ⩾ ⋯ and call the value E ( A ) = σ 1 ( A ) + σ 2 ( A ) + ⋯ the energy of A , thereby extending the concept of graph energy, introduced by Gutman. Let 2 ⩽ m ⩽ n , A be an m × n nonnegative matrix with maximum entry α , and ‖ A ‖ 1 ⩾ n α . Extending previous results of Koolen and Moulton for graphs, we prove that E ( A ) ⩽ ‖ A ‖ 1 m n + ( m − 1 ) ( ‖ A ‖ 2 2 − ‖ A ‖ 1 2 m n ) ⩽ α n ( m + m ) 2 . Furthermore, if A is any nonconstant matrix, then E ( A ) ⩾ σ 1 ( A ) + ‖ A ‖ 2 2 − σ 1 2 ( A ) σ 2 ( A ) . Finally, we note that Wigner's semicircle law implies that E ( G ) = ( 4 3 π + o ( 1 ) ) n 3 / 2 for almost all graphs G .
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Cites
Horn, R. A., & Johnson, C. R. (1985) · Cited by 22,203
Bollobás, B. (1998) · Graduate Texts in Mathematics · doi:10.1007/978-1-4612-0619-4 · Cited by 3,772
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On the Distribution of the Roots of Certain Symmetric Matrices
Wigner, E. P. (1958) · The Annals of Mathematics · doi:10.2307/1970008 · Cited by 1,463
Balakrishnan, R. (2004) · Linear Algebra and its Applications · doi:10.1016/j.laa.2004.02.038 · Cited by 670
The eigenvalues of random symmetric matrices
Füredi, Z., & Komlós, J. (1981) · Combinatorica · doi:10.1007/bf02579329 · Cited by 633
Koolen, J. H., & Moulton, V. (2001) · Advances in Applied Mathematics · doi:10.1006/aama.2000.0705 · Cited by 233
A proof of alon's second eigenvalue conjecture
Friedman, J. (2003) · Proceedings of the thirty-fifth annual ACM symposium on Theory of computing · doi:10.1145/780542.780646 · Cited by 192
Maximal Energy Bipartite Graphs
Koolen, J. H., & Moulton, V. (2003) · Graphs and Combinatorics · doi:10.1007/s00373-002-0487-7 · Cited by 93
Cited by
Huang, C.-H., Tsai, J. J. P., Kurubanjerdjit, N., & Ng, K.-L. (2019) · doi:10.1101/536318 · Cited by 4
Metadata
- Authors
- Nikiforov, V.
- Venue
- Journal of Mathematical Analysis and Applications
- Publisher
- Elsevier BV
- Published
- 2007-02
- Volume
- 326
- Issue
- 2
- Pages
- 1472-1475
- ISSN
- 0022-247X (ISSN-L)
- OpenAlex
- W2090143112
- Cited by
- 334
- References
- 9
- FWCI
- 11.63
- Type
- journal article
- Language
- en
- Open access
- bronze
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