**Author(s):**
Bo Deng*,
Ning Yang,
Weilin Liang,
Xiaoyun Lu

**Journal Name:** Combinatorial Chemistry & High Throughput Screening

Accelerated Technologies for Biotechnology, Bioassays, Medicinal Chemistry and Natural Products Research

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.

**Keywords: **Energy (of a graph), digraphs, adjacency matrix.

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