Date of Award

8-2020

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematics

Thesis Sponsor/Dissertation Chair/Project Chair

Jonathan Cutler

Committee Member

Deepak Bal

Committee Member

Aihua Li

Abstract

In this work, we investigate bounds on the number of independent sets in a graph and its complement, along with the corresponding question for number of dominating sets. Nordhaus and Gaddum gave bounds on χ(G)+χ(G) and χ(G) χ(G), where G is any graph on n vertices and χ(G) is the chromatic number of G. Nordhaus-Gaddum- type inequalities have been studied for many other graph invariants. In this work, we concentrate on i(G), the number of independent sets in G, and ∂(G), the number of dominating sets in G. We focus our attention on Nordhaus-Gaddum-type inequalities over trees on a fixed number of vertices. In particular, we give sharp upper and lower bounds on i(T )+ i(T ) where T is a tree on n vertices, improving bounds and proofs of Hu and Wei. We also give upper and lower bounds on i(G) + i(G) where G is a unicyclic graph on n vertices, again improving a result of Hu and Wei. Lastly, we investigate ∂(T )+ ∂(T ) where T is a tree on n vertices. We use a result of Wagner to give a lower bound and make a conjecture about an upper bound.

File Format

PDF

Download

Included in

Mathematics Commons

Share

COinS