Resolvent Energy of Digraphs

The resolvent energy of a graph G is defined as ER(G) =∑ , where λ ≥ λ ≥ ⋯ ≥ λ 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) = ∑ , where σ ≥ σ ≥ ⋯ ≥ σ are the singular values of G. The second approach, define the resolvent energy of a digraph G by ER(G) = ∑ , n (I ) where z , … , z are the eigenvalues of G and Re(z ) denotes the real part of z . 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.