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Digraph Energy of Directed Polygons

Author(s):

Bo Deng*, Ning Yang, Weilin Liang and Xiaoyun Lu   Pages 1 - 6 ( 6 )

Abstract:


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.

Affiliation:

School of Mathematics and Statistics, Qinghai Normal University, Xining, 810001, School of Mathematics and Statistics, Qinghai Normal University, Xining, 810001, School of Mathematics and Statistics, Qinghai Normal University, Xining, 810001, School of Mathematics and Statistics, Qinghai Normal University, Xining, 810001



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