The Normalized Laplacians, Degree-Kirchhoff Index, and the Complexity of Möbius Graph of Linear Octagonal-Quadrilateral Networks - Jia-Bao Liu; Qian Zheng; Sakander Hayat (Journal of Mathematics)
The normalized Laplacian plays an indispensable role in exploring the structural properties of irregular graphs. Let Ln8,4 represent a linear octagonal-quadrilateral network. Then, by identifying the opposite lateral edges of Ln8,4, we get the corresponding Möbius graph MQn8,4. In this paper, starting from the decomposition theorem of polynomials, we infer that the normalized Laplacian spectrum of MQn8,4 can be determined by the eigenvalues of two symmetric quasi-triangular matrices ℒA and ℒS 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 MQn8,4.