  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 $\tau_m^r(\cdot)$
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.
