#### Date of Award

1-1-2009

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Clark,Lane

#### Abstract

This paper is a compilation of results in combinatorics utilizing the probabilistic method. Below is a brief description of the results highlighted in each chapter. Chapter 1 provides basic definitions, lemmas, and theorems from graph theory, asymptotic analysis, and probability which will be used throughout the paper. Chapter 2 introduces the independent domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the independent domination number is one of two values. Also, the the number of independent dominating sets of given cardinality is analyzed statistically. Chapter 3 introduces the tree domination number. It is then shown that in the random graph model G(n,p) with probability tending to one, the tree domination number is one of two values. Additional related domination parameters are also discussed. Chapter 4 introduces a generalized rook polynomial first studied by J. Goldman et al. Central and local limit theorems are then proven for certain classes of the generalized rook polynomial. Special cases include known central and local limit theorems for the Stirling numbers of the first and second kind and additionally new limit theorems for the Lah numbers and certain classes of known generalized Stirling numbers. Chapter 5 introduces the Kneser Graph. The exact expected value and variance of the distance between [n] and a vertex chosen uniformly at random is given. An asymptotic formula for the expectation is found.

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