The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks

Abstract

The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let $L_n^{\mathrm{8,4}}$ represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of $L_n^{\mathrm{8,4}}$ , we get the corresponding Möbius graph $M{Q}_n\left(\mathrm{8,4}\right)$ . In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of $M{Q}_n\left(\mathrm{8,4}\right)$ can be determined by the eigenvalues of two symmetric quasi-triangular matrices ${\mathrm{ℒ}}_A$ and ${\mathrm{ℒ}}_S$ of order $4n$ . Next, owing to the relationship between the two matrix roots and the coefficients mentioned above, we derive the explicit expressions of the degree-Kirchhoff indices and the complexity of $M{Q}_n\left(\mathrm{8,4}\right)$ .