  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.
