Energy of matrices.

Let M n ( C ) denote the space of n × n matrices with entries in C . We define the energy of A ∈ M n ( C ) as (1) E ( A ) = ∑ k = 1 n | λ k − t r ( A ) n | where λ 1 , … , λ n are the eigenvalues of A, tr ( A ) is the trace of A and | z | denotes the modulus of z ∈ C . If A is the adjacency matrix of a graph G then E ( A ) is precisely the energy of the graph G introduced by Gutman in 1978. In this paper, we compare the energy E with other well-known energies defined over matrices. Then we find upper and lower bounds of 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. [ABSTRACT FROM AUTHOR]