Energy of matrices. (English) Zbl 1426.05054
Summary: Let \(M_n(\mathbb{C})\) denote the space of \(n\times n\) matrices with entries in \(\mathbb{C} \). We define the energy of \(A \in M_n(\mathbb{C})\) as \[\mathcal{E}(A) = \sum_{k = 1}^n \left| \lambda_k - \frac{\operatorname{tr}(A)}{n} \right|\] where \(\lambda_1, \ldots, \lambda_n\) are the eigenvalues of \(A, \operatorname{tr}(A)\) is the trace of \(A\) and \(|z|\) denotes the modulus of \(z \in \mathbb{C} \). If \(A\) is the adjacency matrix of a graph \(G\) then \(\mathcal{E}(A)\) is precisely the energy of the graph \(G\) introduced by I. Gutman [Ber. Math.-Stat. Sekt. Forschungszent. Graz 103, 22 S. (1978; Zbl 0402.05040)]. In this paper, we compare the energy \(\mathcal{E}\) with other well-known energies defined over matrices. Then we find upper and lower bounds of \(\mathcal{E}\) which extend well-known results for the energies of graphs and digraphs. Also, we obtain new results on energies defined over the adjacency, Laplacian and signless Laplacian matrices of digraphs.
MSC:
| 05C20 | Directed graphs (digraphs), tournaments |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
Citations:
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