Asian Research Journal of Mathematics

This study investigates how global spectral characteristics of a coupling network relate to modulation instability (MI) in a nonlinear three-core optical waveguide array. The discrete three-core structure is represented as a weighted graph, enabling the spatial coupling operator to be expressed through an effective directed graph Laplacian. Optical propagation is described by coupled nonlinear Schr¨odinger equations that include unequal nearest-neighbour coupling strengths and Kerr nonlinearities. A continuous-wave background is perturbed by weak sidebands, and linearisation leads to a six-dimensional spectral eigenvalue problem for the perturbation propagation constants. Two graph-based descriptors, the Kirchhoff index and the Laplacian Estrada index, are evaluated to relate the coupling spectrum to the instability response. The analysis shows that variations in the Kirchhoff index are associated with changes in the maximum MI gain and the extent of the unstable region, whereas variations in the Laplacian Estrada index are reflected mainly in the threshold for the onset of instability. Direct numerical integration of the coupled propagation equations produces perturbation growth patterns that are consistent with the linear-stability results for the spectral states considered. The findings indicate that the two indices provide complementary descriptions of instability strength and instability onset in the three-core system, thereby offering a compact graph-spectral framework for relating coupling structure to nonlinear propagation dynamics.