Special Type Of Symbol Mode

The sign pattern matrix is an important and foundational problem in the domain ofcombinatorial mathematics, and its research and future development are widespread. Ithas important application in many subjects such as computer science, economics, physics,chemistry and sociology. In this paper, we firstly study the necessary condition for a spec-trally arbitrary complex sign pattern, and prove a minimal spectrally arbitrary complex signpattern. Then we discuss the bounds on the kth upper bases of primitive non-powerfulanti-symmetric signed digraphs.In chapter 1, we introduce the history of development on the sign pattern matrices,some methods used in our paper and our research problems and main results.In chapter 2, we firstly characterize a necessary condition for a spectrally arbitrary com-plex sign pattern through signed two-colored digraph. Then we find out a minimal spectrallyarbitrary complex sign pattern. Finally we discuss the minimum number of nonzero partsin a spectrally arbitrary complex sign pattern, and give a conjecture.In chapter 3, we obtain the bounds on the kth upper bases of primitive non-powerfulanti-symmetric signed digraphs and prove that there exist some primitive non-powerful anti-symmetric signed digraphs with kth upper bases equal to the bounds.