  In this work, we study a mathematically rigorous metric of a graph
visualization quality under conditions that relate to visualizing a bipartite
graph. Namely we study rectilinear crossing number in a special arrangement of
the complete bipartite graph where the two parts are placed on two concentric
circles. For this purpose, we introduce a combinatorial formulation to count
the number of crossings. We prove a proposition about the rectilinear crossing
number of the complete bipartite graph. Then, we introduce a geometric
optimization problem whose solution gives the optimum radii ratio in the case
that the number of crossings for them is minimized. Later on, we study the
magnitude of change in the number of crossings upon change in the radii of the
circles. In this part, we present and prove a lemma on bounding the changes in
the number of crossings of that is followed by a theorem on asymptotics of the
bounds.
