Spectral Arbitrariness Study Of Special Symbol Patterns

Combinatorial mathematics is an active field of the basic mathematics.In recent years,More and more literatures about combinatorial mathematics are studied at home and abroad.With the development of computer mathematics,combinatorial mathematics has become the important research object in various fields.The sign pattern matrix is the main component in the combinatorial mathematics and it is widely used in various disciplines.The main research direction about sign pattern matrices involves solvability,stability,intertia and the property of the power sequence of a sign pattern matrix.This paper mainly study the sign pattern matrices in the combinatorial mathematics,and prove some classes of the special sign pattern matrixs and its superpatterns are spectrally arbitrary.In chapter 1,we introduce the history and the development of sign pattern matrices at home and abroad,some basic concept and our research problems and main results;In chapter 2,we improve that a class of ray patterns with 3n nonzero entries and its superpatterns are spectrally arbitrary;In chapter 3,we prove the application for the Nilpotent-Jacobi method in complex sign pattern;In chapter 4,we prove that the complex sign pattern and its superpattern are minimal spectrally arbitrary.