Date of Award


Document Type


Degree Name

Master of Science (MS)


College of Science and Mathematics



Thesis Sponsor/Dissertation Chair/Project Chair

Jonathan Cutler

Committee Member

Deepak Bal

Committee Member

Aihua Li


Trees (Graph theory), Domination (Graph theory)


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.

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