Background: The energy E(G) of G is deﬁned as the sum the absolute values of the eigenvalues of its adjacency matrix. In theoretical chemistry, within the Hu ̈ckel molecular orbital (HMO) approximation, the energy levels of the π-electrons in molecules of conjugated hydrocarbons are related to the energy of the molecular graphs.
Objective: Generally, the energy to digraphs was proposed.
Methodology: Let Δ_n be the set consisting of digraphs with n vertices and each cycle having length≡2 mod(4). The set of all the n-order directed hollow k-polygons in Δ_n based on a k-polygon G is denoted by H_k (G).
Results: In this research, by using the quasi-order relation over Δ_n and the characteristic polynomials of digraphs, we describe the directed hollow k-polygon with the maximum digraph energy in H_k (G).
Conclusion: The n-order oriented hollow k-polygon with the maximum digraph energy among Hk(G) only contains a cycle. Moreover, such a cycle is the longest one produced in G.