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Independence polynomials of sparse graphs
Author
Ng, Boon Leong
Supervisor
Dong, F. M.
Abstract
This thesis covers the independence polynomials of two classes of sparse graphs, namely, paths and cycles. Chapter 1 covers the history of the study of independence polynomials, with a particular emphasis on investigations into independence equivalence in general. Relevant results on the factorisation and roots of the independence polynomials of paths and cycles, as well as past results on the independence equivalence
classes of paths and cycles are also mentioned and proven here. The author’s results, Theorems 1.6.4 and 1.6.10, are also stated here.
Chapters 2 and 3 are devoted to finding the independence equivalence classes of cycles and paths respectively. Chapter 2 is a proof of Theorem 1.6.4 and Chapter 3 is a proof of Theorem 1.6.10. These are both long proofs due to the number of subcases involved, and the fact that (particularly for Theorem 1.6.10) each of these subcases needs to be handled in its own idiosyncratic manner. These two chapters extend work done by Beaton, Brown and Cameron [7].
Finally, Chapter 4 is an exploration into the independence fractals of paths and cycles, and an attempt to find a bounding set for these independence fractals. Counterexamples to a conjecture made by Alikhani and Peng [3] are presented here and a method for constructing a bounding set for the independence fractals of paths and cycles is described.
classes of paths and cycles are also mentioned and proven here. The author’s results, Theorems 1.6.4 and 1.6.10, are also stated here.
Chapters 2 and 3 are devoted to finding the independence equivalence classes of cycles and paths respectively. Chapter 2 is a proof of Theorem 1.6.4 and Chapter 3 is a proof of Theorem 1.6.10. These are both long proofs due to the number of subcases involved, and the fact that (particularly for Theorem 1.6.10) each of these subcases needs to be handled in its own idiosyncratic manner. These two chapters extend work done by Beaton, Brown and Cameron [7].
Finally, Chapter 4 is an exploration into the independence fractals of paths and cycles, and an attempt to find a bounding set for these independence fractals. Counterexamples to a conjecture made by Alikhani and Peng [3] are presented here and a method for constructing a bounding set for the independence fractals of paths and cycles is described.
Date Issued
2022
Call Number
QA166.22 Ng
Date Submitted
2022