On the number of spanning trees and normalized Laplacian of linear octagonal‐quadrilateral networks

The normalized Laplacian makes a great contribution on analyzing the structure properties of nonregular graphs. Let On be a linear octagonal‐quadrilateral network. In this article, we first concern the normalized Laplacian spectrum of On based on the decomposition theorem for the corresponding matrices. Then we derive the closed‐term formulas of the degree‐Kirchhoff index and the number of spanning trees of linear octagonal‐quadrilateral networks on the basis of the relations between the roots and coefficients, respectively. The degree‐Kirchhoff index and complexity are the important parameters to explore the structural properties of a given network. The normalized Laplacian plays a key role on studying the structure properties of nonregular graphs. The formulas of the degree‐Kirchhoff index and the exactly complexity of the linear octagonal‐quadrilateral graphs are given in terms of the normalized Laplacian.