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Spectral Analysis and its applications for a class of scale-free network based on the weighted m-clique annex operation

The spectrum of network is an important tool to study the function and dynamic properties of network, and graph operation and product is an effective mechanism to construct a specific local and global topological structure. In this study, a class of weighted $m-$clique annex operation $τ_m^r(\cdot)$ controlled by scale factor $m$ and weight factor $r$ is defined, through which an iterative weighted network model $G_t$ with small-world and scale-free properties is constructed. In particular, when the number of iterations $t$ tends to infinity, the network has transfinite fractal property. Then, through the iterative features of the network structure, the iterative relationship of the eigenvalues of the normalized Laplacian matrix corresponding to the network is studied. Accordingly, some applications of the spectrum of the network, including the Kenemy constant, Multiplicative Degree-Kirchhoff index and the number of weighted spanning trees, are further given. In addition, we also study the effect of the two factors controlling network operation on the structure and function of the iterative weighted network $G_t$, so that the network operation can better simulate the real network and have more application potential in the field of artificial network.