The energy of directed hexagonal systems

The energy of a digraph D is defined as E(D)=∑i=1n|Re(zi)|, where Re(zi) denotes the real part of the complex number zi. We study in this work the energy over the set Δn consisting of digraphs with n vertices and cycles of length ≡2mod(4). Due to the fact that the characteristic polynomial of a digraph D∈ Δn has an expression of the formΦD(z)=zn+∑k=1⌊n2⌋(−1)kc2k(D)zn−2k where c2k(D) are nonnegative integers, it is possible to define a quasi-order relation over Δn, in such a way that the energy is increasing. Moreover, we show that the energy of a digraph D∈Δn decreases when an arc of a cycle of length 2 is deleted. Consequently, we obtain extremal values of the energy over sets of directed hexagonal systems, i.e. digraphs whose underlying graph is a hexagonal system.