Adjacency-distance and generalized adjacency distance energies of non-commuting graph for dihedral groups

The correspondence between matrix and graph is one of the fundamental characteristics of the spectral graph theory. This relation allows us to formulate the characteristic polynomial of the graph and calculate the energy of the graph as the sum of its absolute eigenvalues. This study examines the non-commuting graph for dihedral groups. We establish the spectral radius and energy of this graph using the adjacency-distance and generalized adjacency-distance matrices. It should be noted that the obtained energies are never an odd integer and are always equal to twice its spectral radius.

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