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The topological properties of the networks can be described by the Laplacian spectra, but resolving the Laplacian spectra of networks poses difficulties. In this study, a novel approach for solving the Laplacian spectrum of weighted composite networks is presented. We first give the definitions of three weighted graph operations, namely, Cartesian product, corona, and join. Second, the Laplacian spectra of these composite networks are calculated. Finally, we use the obtained Laplacian spectrum to deduce some topological properties of the networks, such as network coherence, entire mean first-passage time, and Laplacian energy, which have several applications in physical chemistry.