The normalized Laplacian spectrum and complexity of the chained interconnection-networks

In this paper, we present a spectral analysis of the strong prism of the chained polyomino network based on its normalized Laplacian matrix. First, by the decomposition theorem for normalized Laplacian characteristic polynomials in product graphs, we partition the normalized Laplacian matrix of the network into two symmetric matrices LA and LS, explicitly deriving their complete eigenvalue sets. Second, we explore the interconnection between the roots and polynomial terms of the characteristic polynomials of these two matrices. Third, the multiplicative degree-Kirchhoff index for chained networks was derived, which analyzes the properties of the network and is especially important to understand the electrical properties of the network. Finally, the complexity of chained networks was derived. These findings establish a solid foundation for understanding the structural complexity and connectivity of networks.