Since 1952, several well-known graph theorists have proven numerous results regarding Hamiltonian graphs. In fact, many elementary graph theory textbooks contain the theorems of Ore, Bondy and Chvatal, Chvatal and Erdos, Posa, and Dirac, to name a few. In this note, the authors state and prove some propositions of their own concerning Hamiltonian graphs. The results are rather elementary and, for that reason, may help beginning graph theory students more fully understand just what makes a graph Hamiltonian. (Contains 4 figures.)