Three Kinds Of Special Research Of Sign Pattern Matrix Spectrum

The sign pattern matrix is a very active research direction in the domain of combi-natorial mathematics in recent years,as an interdiscipline of linear algebra, combinatorialmathematics and graph theory, it has important application in many subjects. In this pa-per, we mainly use the Nilpotent-Jacobian method to prove that two classes of sign patternsmatrix and a complex sign pattern matrix are spectrally arbitrary.In chapter 1,we introduce the history of combinatorial mathematics and sign patternmatrices, some methods used in our paper and our research problems and main results.In chapter 2,we prove a class of sign pattern matrix of order n≥4 is a minimallyspectrally arbitrary sign pattern,and every superpattern of it is a spectrally arbitrary signpattern.In chapter 3,we prove a class of sign pattern matrix of order n≥3 is a minimallyspectrally arbitrary sign pattern,and every superpattern of it is a spectrally arbitrary signpattern.In chapter 4,we used the Nilpotent-Jacobian method to prove a complex sign patternmatrix is a spectrally arbitrary sign pattern,and every superpattern of it is a spectrallyarbitrary sign pattern.