<![CDATA[Graphical representations of algebraic structures have become an important tool in modern mathematics and its applications. Graph theory, particularly spectral graph theory, is widely used across disciplines such as chemistry, physics, computer science, and network analysis to study structural and functional relationships. This study focuses on the non-coprime graph of the dihedral group \( D_{2n} \) with \(n=p^k \), where $p$ is a prime number, and \(k \in \mathbb{Z}^{+}\), analyzing two fundamental spectral parameters: the Estrada index and the Laplacian Estrada index, which are defined based on the eigenvalues of the graph’s adjacency and Laplacian matrices. The main result of this research is the derivation of explicit general formulas for both indices on the non-coprime graph of the dihedral group, contributing to the advancement of algebraic graph theory through spectral analysis.]]>
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