Computing the Laplacian spectrum and Wiener index of pentagonal-derivation cylinder/Möbius network

The Laplacian spectrum significantly contributes the study of the structural features of non-regular networks. Actually, it emphasizes the interaction among the network eigenvalues and their structural properties. Let $$$$ represent the pentagonal-derivation cylinder (Möbius) network. In this article, based on the decomposition techniques of the Laplacian characteristic polynomial, we initially determine that the Laplacian spectra of $$ contain the eigenvalues of matrices $$ and $$. Furthermore, using the relationship among the coefficients and roots of these two matrices, explicit calculations of the Kirchhoff index and spanning trees of $$ are determined. The relationship between the Wiener and Kirchhoff indices of $$ is also established.