Energy of strongly connected digraphs whose underlying graph is a cycle

Abstract

The energy of a digraph is defined as E (D) =∑₁n|Re (z_k)|, where z₁,..., z_nare the eigenvalues of the adjacency matrix of D. This is a generalization of the concept of energy introduced by I. Gutman in 1978 [3]. When the characteristic polynomial of a digraph D is of the form

where bo (D) = 1 and b_k(D) ≥ 0 for all k, we show that

This expression for the energy has many applications in the study of extremal values of the energy in special classes of digraphs. In this paper we consider the set D* (C_n) of all strongly connected digraphs whose underlying graph is the cycle C_n, and characterize those whose characteristic polynomial is ofthe form (0.1). As a consequence, we find the extremal values ofthe energy based on (0.2).