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This research focuses on analyzing the energy and Laplacian energy of the relative coprime
graphs of symmetric groups for degrees three and four to enhance understanding of their algebraic
and spectral properties through graphical representations. The objectives are to construct the
relative coprime graphs for S3 and S4, compute their energy and Laplacian energy, and compare
the results to uncover significant patterns or differences. Addressing the limited research on
the relative coprime graphs of symmetric groups, this study aims to fill the gap by providing a
detailed analysis of these graphs. The methodology involves constructing the graphs by listing
all elements of the symmetric groups, determining their orders, and drawing edges between
vertices that are relatively prime. The energy of a graph is calculated by determining the
eigenvalues of its adjacency matrix and summing their absolute values, while the Laplacian
energy is derived from the eigenvalues of the graph’s Laplacian matrix. The results show
distinct numerical differences and similarities between S3 and S4, with S3 exhibiting relatively
close Laplacian energy values across subgroups and S4 showing a wider variance, indicating
sensitivity to the number of elements and subgroup complexity. These findings contribute
valuable insights into the spectral characteristics and structural resilience of symmetric groups,
providing a foundation for future research on more complex groups and their applications in
various scientific domains