Digraph Energy of Directed Polygons.

Area of Science:

• Graph theory
• Theoretical chemistry
• Mathematical chemistry

Background:

• Graph energy E(G) is the sum of absolute eigenvalues of the adjacency matrix.
• Huckel molecular orbital (HMO) approximation relates π-electron energy levels to molecular graph energy.

Purpose of the Study:

• To find the digraph with maximum digraph energy within a defined class of graphs.
• To characterize the structure of such a digraph.

Main Methods:

• Definition of the set Δn comprising digraphs with n vertices and cycles of length 2 mod(4).
• Introduction of the set Hk(G) for n-order directed hollow k-polygons.
• Utilizing quasi-order relations and characteristic polynomials of digraphs.

Main Results:

• The directed hollow k-polygon with maximum digraph energy in Hk(G) was described.
• This polygon was found to contain a single cycle.

Conclusions:

• The n-order oriented hollow k-polygon with maximum digraph energy within Hk(G) consists solely of a cycle.
• This cycle is the longest cycle present in the graph G.