The energy of graphs and matrices. (English) Zbl 1113.15016

The author defines the energy \({\mathcal E}(A)\) of a complex rectangular matrix \(A\) as the sum of its singular values \(\sigma_1(A)\geq ...\geq \sigma_{m\wedge n}(A),\) extending hereby the concept of the energy of a graph \(G\) (defined via the adjacency matrix \(A(G)\)); see the review cited below). He shows for any nonconstant matrix \(A\) that \(\sigma_1(A)+\left(| | A| | _2^2-\sigma_1^2(A)\big/ \sigma_2(A)\right)\leq {\mathcal E}(A), \) while for a nonnegative \(m\times n\) matrix with maximum entry \(\alpha\) and \(| | A| | _1\geq n\alpha,\) there holds
\[ {\mathcal E}(A)\leq \frac{| | A| | _1}{\sqrt{mn}}+ \sqrt{(m-1)\left( | | A| | _2^2-\frac{| | A| | _1^2}{mn} \right)} \leq \alpha \frac{\sqrt{n}(m+\sqrt{n})}{2}. \]
For matrices \(A=A(G)\) this was found earlier by J. H. Koolen and V. Moulton [Adv. Appl. Math. 26, 47–52 (2001; Zbl 0976.05040)]. Using Wigner’s semicircle law he also gets for almost all graphs \(G\) that \({\mathcal E}(A(G))=\left(\frac{4\pi}{3}+o(1)\right)n^{3/2}.\)
There are no hints to that the author has consulted any of the numerous articles on estimates for subsums of eigenvalues, see e.g. J. K. Merikoski and A. Virtanen [Linear Algebra Appl. 264, 101–108 (1997; Zbl 0885.15011)], where very similar formulae can be found.


15A42 Inequalities involving eigenvalues and eigenvectors
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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