Analysis of Approximation Methods of Laplacian Eigenvectors of the Kronecker Product of Graphs

This paper analyzes two approximation methods for the Laplacian eigenvectors of the Kronecker product, as recently presented in the literature. We enhance the approximations by comparing the correlation coefficients of the eigenvectors, which indicate how well an arbitrary vector approximates a matrix’s eigenvector. In the first method, some correlation coefficients are explicitly calculable, while others are not. In the second method, only certain coefficients can be estimated with good accuracy, as supported by empirical and theoretical evidence, with the rest remaining incalculable. The primary objective is to evaluate the accuracy of the approximation methods by analyzing and comparing limited sets of coefficients on one hand and the estimation on the other. Therefore, we compute the extreme values of the mentioned sets and theoretically compare them. Our observations indicate that, in most cases, the relationship between the majority of the values in the first set and those in the second set reflects the relationship between the remaining coefficients of both approximations. Moreover, it can be observed that each of the sets generally contains smaller values compared to the values found among the remaining correlation coefficients. Finally, we find that the performance of the two approximation methods is significantly influenced by imbalanced graph structures, exemplified by a class of almost regular graphs discussed in the paper.