The Generalized Laplace Polynomial Of The Network

In this paper,Laplace polynomials and their spectrum in complex networks are studied.Since Laplace spectrum theory is a very powerful tool for studying complex networks,the knowledge of Laplace spectrum can be used to explore the evolution process of network model and its topological structure.The research on the controlling of the network,Alexander et al.have studied the relationship between the U-controllable graph and the main eigenvalues of corresponding matrix.Its important research result lies in the generalization of Laplace matrix.But relatively speaking,some basic results have not been obtained in the study of this generalized Laplace matrix.In this paper,a class of generalized Laplace polynomials and the corresponding characteristic spectra of a network are studied.The main contents are as follows:1.The generalized Laplace polynomials and characteristic spectrum of the global-coupled network and the complete two division network are studied.In addition,two important corollaries are given for the complete two division network.2.The method of calculating the generalized Laplace polynomial of network with edge-cut is presented and its recursive algorithm is given.