Digraph Energy of Directed Polygons

Background: The energy E(G)of Graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In theoretical chemistry, within the Huckel 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: The digraph with maximum digraph energy in a class of graphs is found. Methodology: Let Δn be the set consisting of digraphs with n vertices and each cycle having length = 2mod(4). The set of all the n-order directed hollow k-polygons in Δn based on a - polygon G is denoted by Hk(G). Results: In this research, by using the quasi-order relation over Δn and the characteristic polynomials of digraphs, we describe the directed hollow k-polygon with the maximum digraph energy in Hk(G). Conclusion: The n-order oriented hollow k-polygon with the maximum digraph energy among H_k (G) only contains a cycle. Moreover, such a cycle is the longest one produced in G.