This analysis reveals the eigenvalues of representation matrices in coprime graphs, suggesting structural insights into the generalized quaternion group.
This study discusses the representation matrices of coprime graph of generalized quaternion group. The representation matrices are adjacency matrix, antiadjacency matrix, Laplacian matrix, and signless Laplacian matrix. Furthermore, the eigenvalues of each representation matrices are determined. As the results, we obtained the construction of the four representation matrices and their eigenvalues. Based on the matrix form, we got the matrix determinant is zero, so that the matrices have zero eigenvalues except signless Laplacian matrix. As for the non-zero eigenvalues, the values depends on the type of representation matrices and the order of the graph as well as its algebraic multiplicity.
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Zahidah et al. (2025) studied this question.
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