Analyzing the normalized Laplacian spectrum and spanning tree of the cross of the derivative of linear networks

In this paper, we focus on the strong product of the pentagonal networks. Let $ R_{n} $ be a hexagonal network composed of $ 2n $ pentagons and $ n $ quadrilaterals. Let $ P_{n}^{2} $ denote the graph formed by the strong product of $ R_{n} $ and its copy $ R_{n}^{\prime} $. By utilizing the decomposition theorem of the normalized Laplacian characteristics polynomial, we characterize the explicit formula of the multiplicative degree-Kirchhoff index completely. Moreover, the complexity of $ P_{n}^{2} $ is determined.