Exponents of Primitive Companion Matrices | Shiv Nadar University
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Exponents of Primitive Companion Matrices

This thesis contains five chapters and is devoted to the study of the exponents of primitive companion matrices and primitive symmetric companion matrices. The first chapter of this thesis is a revision of basic definitions and results of matrix theory and graph theory. A literature survey on the study of the exponents of primitive matrices can be found in the second chapter. The third chapter of the thesis builds on the work of the study of F_n (x, k), the number of binary strings of length n containing x zeros and a longest subword of k zeros. In this chapter, we establish a few results on F_n (x, k) and find some applications of it. The fourth chapter contains the total number of primitive and the total number of imprimitive symmetric companion matrices. Also, we obtain formulas to compute the exponent of every primitive symmetric companion matrix. Finding the number of primitive symmetric companion matrices with a given exponent for certain cases is also a part of this chapter. In the fifth chapter, we find the number of primitive (0, 1) companion matrices of order n. If X denotes the set of all such matrices, then we explore the exponent set E(X), where E(X) = {m ∈ N : there exists an n × n matrix A in X with exp(A) = m}. We also ask a few questions according to the context of these chapters, which may give impetus to the reader to work further in this direction.

Student Name: 
Monimala Nej
Faculty Advisor: