Fixed Number Of Groups With A Directed Graph Digraph Of Scrambling Index

In the field of combinatorial matrix theory, the research on the exponent of non-negativematrix and sign pattern matrix is an important and practical issue. It has specific application inmany fields like information science, computer science and the information transfer etc.Graph theory plays an important role in the research of combinatorial matrix theory, as thegraph-theoretical methods are often useful in the study of the powers of matrices. By studyingthe associated digraph of non-negative matrix or the associated signed digraph of sign patternmatrix, we can solve the problem of matrix theory. In this work, the local exponents and thescrambling indices of several classes of digraph and the local bases of several classes ofsigned digraph are studied.In the first chapter, some concepts and background of the combinatorial matrix theoryand graph theory are introduced. The concepts of primitive digraph, primitive non-powerfuldigraph, exponent, scrambling index and base are introduced also. At the same time, the mainconclusion is given.In the second chapter, two classes of primitive digraphs that contain three circles arestudied. According to the definition of exponent and using the reduction to absurdity, weobtain the local exponents of them.In the third chapter, two classes of primitive non-powerful signed digraphs that containthree circles are studied. By discussing whether there are special SSSD walks and by usingthe reduction to absurdity, we obtain the local bases of them.In the last chapter, a special primitive digraph D that contains three circles is studied.Let v is a vertex of D. By analyzing the set of vertex which can be reached through a tlength walk from the vertex v, we get the scrambling index of the digraph.