First, I introduce quantum graph theory. I also discuss a known lower bound on the independence numbers and derive from it an upper bound on the chromatic numbers of quantum graphs. Then, I construct a family of quantum graphs that can be described as tropical, cyclical, and commutative. I also define a step logarithm function and express with it the bounds on quantum graph invariants in closed form. Finally, I obtain an upper bound on the independence numbers and a lower bound on the chromatic numbers of the constructed quantum graphs that are similar in form to the existing bounds. I also show that the constructed family contains graphs of any valence with arbitrarily high chromatic numbers and conclude by it that quantum graph colorings are quite different from classical graph colorings.