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

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