The Minimum Degree Energy of the Cayley Graph Associated to the Dihedral Group of Order Six with Subsets of Order Two and Three

Abstract

The energy of a simple graph in graph theory is defined as the sum of the absolute values of the eigenvalues of the graph's adjacency matrix, a concept inspired by Hückel Molecular Orbital theory. Chemists originally used this idea to estimate the energy associated with π-electron orbitals in conjugated hydrocarbons. The minimum degree energy, on the other hand, is defined as the sum of the absolute values of the eigenvalues of the graph's minimum degree matrix. A Cayley graph associated to a finite group with a subset  is defined as a graph in which the vertices are the elements of the group and two vertices  and  are joined with an edge if and only if  is equal to the product of and  for some elements  in the subset . In this research, we compute the minimum degree energy of Cayley graphs associated with the dihedral group of order six, focusing on subsets of orders two and three. The process involves constructing the Cayley graph for each subset, determining the minimum degree matrix, and calculating the corresponding eigenvalues. The findings indicate that for subsets of order two, the minimum degree energy is 16, while for subsets of order three, the minimum degree energies is 18 or 24. Notably, the minimum degree energy is an even number for all cases.