Primitive Index And Spectrally Arbitrary Pattern

Graph theory and Sign pattern matrix are very active research topic in combinatorialmathematics,and one of the important reasons is that it has wide application in many sub-jects such as economics, biology,physics, chemistry, operation research, computer science,information theory, cybernetics, network theory, social science. In this paper, we consider aclass of primitive two-colored digraphs with two cycles, and prove two sign patterns whichare spectrally arbitrary sign patterns with the tools of N-J .In chapter 1, firstly, we introduce the history of development on graph theory and signpattern matrices. Secondly, we introduce some elementary knowledge and research surveyabout the primitive exponents of directed digraph and spectrally arbitrary sign pattern.Lastly, we propose our research problems.In chapter 2, we consider the special two-colored digraphs D who has 2n - 3 verticesand consists of one n-cycle and one (n-2)-cycle. Some primitive conditions are proved. Wefind the bounds on the exponents. Further, we prove that the bonds of extremal two-coloreddigraphs can be reached.In chapter 3 and Chapter 4, we separately discuss that P is a spectrally arbitrary signpattern and Q is a minimally spectrally arbitrary sign pattern, and every super-pattern ofit is a spectrally arbitrary sign pattern.