A New Notion of Energy of Digraphs

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. Let z1(. . . zn be the eigenvalues of an n-vertex digraph D. Here, we give a new notion of energy of digraphs defined by Ep (D ) =∑^n k = 1| R (zk) 3 (zk) | , where R (zk) and S (zk) are real and imaginary parts of zk, respectively. We call it pP-energy of the digraph and compute the P-energy formulas for directed cycles. For n 1 2 , it is shown that pP-energy of directed cycles increases monotonically with respect to their order. The unicyclic digraphs with smallest and largest P-energy are obtained and counter examples will be presented to show that the P-energy of a digraph does not possess increasing-property with respect to quasi-order relation over the set , where is the set of n-vertex digraphs with cycles of length . Also, an upper bound for the P-energy is presented and give all digraphs which attain this bound.