Geometric and Algebraic Graphs and their Applications

In this thesis, we begin with a look at the Wiener Index and its correlation with the boiling point of a hydrocarbon. Though the Wiener Index correlates physical properties of chemical compounds relatively well, we produce a new method of modeling the hydrocarbons and their chemical property which creates a geometric graph that we can analyze. Then we take a look at a classic conjecture colloquially known as The Middle Two Levels Conjecture. This problem in pure graph theory involves finding Hamiltonian cycles in a power set on n elements. We take advantage of the fact that it is an algebraic graph to produce a method of finding large cycles in the graph. Finally we investigate a method of finding Super Edge-graceful labeling (SEGL) on graphs. We make an effort to generate infinite families that have a SEGL and continue research to be able to classify graphs into categories of those with a SEGL and those without