Three Kinds Of Minimal Spectral Arbitrary Patterns With 2n Nonzero Metals

The theory of sign pattern matrices is an important branch in the research area ofcombinatorial 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, biology, computer science and so on.This thesis will study three classes of special spectrally arbitrary sign pattern matrices byusing nilpotent-jacobian method. The outline of this dissertation is as follows.In chapter1, we introduce the history and meaning of researches on combinatorialmathematics. The related conceptions of sign pattern matrices are also presented.In chapter2, three methods are presented, which can be used to prove sign patternmatrices are spectrally arbitrary, they are construction method, nilpotent-jacobian method andnilpotent-centralizer method.In chapter3, by using nilpotent-jacobian method, two spectrally arbitrary sign patternmatrices are presented. Furthermore, we proved they are also minimal spectrally arbitrary.In chapter4, a special spectrally arbitrary sign pattern matrix is presented. In themeantime, we proved that it is also minimal spectrally.