The Laplacian spectra of the RG-join weighted graphs and related asymptotic network indices

In this article, the Laplacian spectra of the RG-join weighted graphs and the related network indices named network coherence, kirchhoff index, and Laplacian-energy-like invariant are studied via algebraic graph theory and analysis approach. First, the Laplacian spectrum of the weighted RG-join graph is derived, then the union of graphs together with the RG-join operation are applied to form the weighted RG-join graphs with several classic substructures, and the mathematical characterizations for the indices are derived by the L-spectra. In addition, the related asymptotic results are also derived. It is found that, based on the RG-join weighted structure, when the cardinalities of all copy sets inner $G_2$ are large enough, the network coherence and the Kirchhoff index will be irrelevant with the quantity of copies in $G_2$, and also irrelevant with the edge weight $d_1$ in the other subgraph $G_1$.