The Scrambling Index Of Some Primitive Matrices

Combinatorial matrix theory is an important field of combinatorial mathematics.It hasclose contaction with number theory, graph theory, linear algebra and probability andstatistics and other branches of mathematics, otherwise it has been widely applied incommunication network theory, sociology, computer science, biology and economics.In2009, Akelbek and Kirkland introduced a parameter called the scrambling index ofprimitive matrix(digraph) from the second largest modulus of eigenvalues of a stochasticmatrix. The scrambling index is a better characterization for relation between matrix arrays,which become a new hotspot after the primitive exponent and the generalized exponent. Inthis work we will study the scrambling index of primitive matrix by using the graphtheory.The major contents as following: In the first chapter we mainly explain some basicconcepts, the research background of the scrambling index and the current researchsituations at home and abroad; In the second chapter we discuss the scrambling index ofzero-trace symmetric primitive matrices; In the third chapter we characterize completely thescrambling index of nonzero-trace symmetric primitive matrices; In the fourth chapter westudy the scrambling index of primitive digraph with Hamilton cycle; In the fifth chapter wepoint out the contents that will be futher studied.