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Digraph Energy of Directed Polygons.

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Summary

Researchers identified the digraph with maximum energy in a specific class of graphs. The study found that the n-order oriented hollow k-polygon with maximum digraph energy contains only a single, longest cycle.

Keywords:
Energy (of a graph); digraphs; adjacency matrixHMOdiagonal matrix.quasi-order

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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.