Sign Pattern Of Primitive Non-power Base With The Spectrum Of An Arbitrary Sign Pattern

Combinatorial matrix theory is an important branch of combinatorial mathematics.Itsmain research content is combinatorial nature of matrix relating with the symbol of ma-trix elements only and Irrelevant with the size of matrix elements.The combinatorial na-ture is very close to some nature of graphs with specific application in information sci-ence,communication network,computer sicence,and so on.In this paper,we study thebases of primitive non-powerful sign patterns and spectrally arbitrary patterns,which areall important research topics in combinatorial matrix theory.In chapter 1,we outlines the history of combinatorial mathematics , the meanings ofsign pattern matrix and spectrally arbitrary and the main work.In chapter 2, we study the bases for a special class of primitive non-powerful signeddigraphs, through the analysis of the primitive non-powerful signed digraphs, we obtain thebounds of primitive non-powerful sign patterns bases .In chapter 3,a minimal spectrally arbitrary sign patterns of order n with 2n nonzeroentries is given by using the Nilpotent-Jacobian method.