resolvent energy of digraphs

The resolvent energy of a graph g is defined as er(g) =∑^n i=1/n-λi, where λ1 ≥ λ2 ≥ ⋯ ≥ λn are the eigenvalues of the adjacency matrix of g. we extend this concept to directed graphs with two approaches. the first approach, consider er(g) =∑^n i =1 1/n-σi where σ1≥ σ2≥...≥σn are the singular values of g. the second approach, define the resolvent energy of a digraph g by er(g) = ∑^n i =1 1/n-r (zi), where z1, … , zn are the eigenvalues of g and re(zi) denotes the real part of zi. we prove some properties of resolvent energy for some special digraphs and determine the resolvent energy of unicyclic and bicyclic digraphs and present lower bound for resolvent energy of directed cycles.