The Two Methods In The Study Of The Spectrum Arbitrary Pattern Matrix

The qualitative theories of sign pattern matrices is an important branch in the researcharea of combinatorial mathematics. The earliest study for sign pattern matrices theories is ineconomics. It has many important applications in mathematics. Meanwhile, a series of itsresearch results also widely apply in economics, computer science, biology and so on.This thesis mainly use two methods to study some classes of special spectrally arbitrarysign pattern matrices. The outline of this dissertation is as follows.Chapter1introduces the study history and meaning of combinatorial mathematics andgraph theories. The related conceptions and relationships of sign pattern matrices are alsopresented.Chapter2presents three method, i.e. construction method, nilpotent-jacobian methodand nilpotent-centralizer method. All of them can prove that sign pattern matrices arespectrally arbitrary.Chapter3presents two special sign pattern matrices and proves they are spectrallyarbitrary sign pattern matrices by using two methods. Finally, we identify that they are alsominimal spectrally arbitrary.Chapter4presents a special sign pattern matrix and proves that it is spectrally arbitraryby using two methods. In the meantime, we also certify that it is minimal spectrally arbitrary.