The planarity, radius, diameter, and degree of the coprime graph of group of units modulo n.

Graph ΓG = G(V, E) of some group G is called coprime, if V = G is finite and E = {gcd (|a|, |b|) = 1, a ≠ b}. Let G be group of units modulo n or U (n), the group of all positive integers less than n and relatively prime with n with respect to the modulo n multiplication. This article studies the properties of the coprime graph of group G = U (n), such as their planarity, radius, diameter, and degree. Therefore, we analyzed the properties of several examples of coprime graphs over some U(n) groups, then made conjectures, and proved it. In addition, it is also found that the coprime graph ΓU(n) is a complete bipartite graph and planar if and only if the U (n) group has order 2m for some positive integer m. [ABSTRACT FROM AUTHOR]