Representation Matrices of Coprime Graph of Generalized Quaternion Group

Abstract

This study discusses the representation matrices of the coprime graph of the generalized quaternion group. The representation matrices are adjacency matrix, anti adjacency matrix, Laplacian matrix, and signless Laplacian matrix. Furthermore, the eigenvalues of each representation matrix are determined. As a result, we obtained the construction of the four representation matrices and their eigenvalues. The matrix determinant is zero based on the matrix form, so the matrices have zero eigenvalues except for the signless Laplacian matrix. As for the non-zero eigenvalues, the values depend on the type of representation matrices, the order of the graph, and its algebraic multiplicity.