Complex adjacency matrix and energy of digraphs

Abstract

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. If D is a digraph of order n with eigenvalues , then its energy is defined by . We give a new notion of energy of digraphs defined by and call it the iota energy of the digraph D. It is shown that the Coulson’s integral formula remains valid for iota energy. We also find the unicyclic digraphs with extremal iota energies among the class of unicyclic digraphs with a fixed order. Furthermore, it is shown that the iota energy is increasing over the set of n-vertex digraphs with cycles of length h, with respect to a quasi-order relation. We also generalize the increasing property of the energy over the set with respect to quasi-order relation.