Virtual Booth

This study investigates the Laplacian energy of relative co-prime graphs associated with dihedral groups of both odd and even degrees less than or equal to 10. The objectives of this study are to determine the Laplacian matrices, derive the characteristic polynomials and eigenvalues, and compute the Laplacian energy of the constructed graphs. The relative co-prime graphs were adapted from previous studies, followed by the construction of the corresponding Laplacian matrices. The characteristic polynomials and eigenvalues are obtained using computational tools, particularly dCode, to facilitate spectral analysis. The findings reveal that the Laplacian energy varies according to the structural complexity, order, and subgroup configurations of the dihedral groups. In general, larger dihedral groups and more complex subgroup structures tend to exhibit higher Laplacian energy values, whereas smaller groups demonstrate lower energy levels. Furthermore, observable patterns in the spectral behaviour indicate a strong relationship between algebraic group structures and graph-theoretic properties. This study contributes to the advancement of algebraic graph theory by extending the understanding of spectral invariants of relative co-prime graphs derived from dihedral groups. The findings may serve as a foundation for future investigations involving other graph energies, such as Seidel energy, or broader classes of algebraic structures.