The Normalized Laplacian Spectrum Analysis of Fractal Mobius Octagonal Networks and its Applications

TL;DR

This paper analyzes the normalized Laplacian spectrum of fractal M"{o}bius octagonal networks, deriving spectral properties and applying them to compute topological indices like spanning trees and Kirchhoff index.

Contribution

It determines the normalized Laplacian spectrum of fractal M"{o}bius octagonal networks and derives formulas for key topological indices based on spectral analysis.

Findings

Spectrum of $Q_n$ expressed via matrices $\\mathcal{L}_A$ and $\\mathcal{L}_S$

Formulas for multiplicative degree-Kirchhoff index

Number of spanning trees for $Q_n$

Abstract

The study and calculation of spectrum of networks can be used to describe networks structure and quantify analysis of networks performance. The fractal M\"{o}bius octagonal networks, denoted by $Q_n$, is derived from the inverse identification of the opposite lateral edges of fractal linear octagonal networks. In this paper, the normalized Laplacian spectrum of $Q_n$ is determined by two matrices $\mathcal {L}_A$ and $\mathcal {L}_S$. As an important application of our results, some topological indices (multiplicative degree-Kirchhoff index, the number of spanning trees) formulas of $Q_n$ are obtained.