Recent work has characterized properties of algebraic structures by exploring their associated directed graphs, which can be thought of as a visual representation of the structure’s Cayley tables. Our work expanded on previous efforts regarding the directed graph of the finite tropical semiring. Building on the previous analysis of the structure of the directed graph, we explore graph-theoretic implications of the minimum and maximum definitions of tropical addition. We give formulas for the in-degree of vertices in the directed graphs and the number of vertices with a given in-degree. We also analyze the connected components of the directed graphs and present formulas for the number of connected components and the greatest common divisor of the vertices in a given component. If time allows, we also discuss the cycles in connected components along with the lengths of paths in a given directed graph.