Applications of Laplacian spectrum for the vertex-vertex graph

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex-vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex-vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like.