Graph Theory can be applied in a nearly innumerable amount of ways as graphs can be representative of almost anything. Graphs themselves, have a large amount of characteristics that can be explored some of which through vertex colorings and simple investigations. In this paper, we will be looking the various behaviors of graphs, the process of vertex coloring, and finally we will construct our own graph. From the construction of this graph, we will further study vertex colorings as well as derive an infinite graph. Our goal is to better understand the basic properties of family graphs and to understand how they behave in relation to vertex colorings.