On the Independence Number of the Prime-Coprime Graph of a Finite Group

The prime-coprime graph $\mathrm{\Theta }(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all finite groups $G$ for which $\mathrm{\Theta }(G)$ is a split graph. We establish a general lower bound for the independence number of $\mathrm{\Theta }(G)$ of an arbitrary finite group $G$. Moreover, we explicitly compute the independence number of $\mathrm{\Theta }(G)$ for several distinguished families of finite groups, including cyclic, dihedral, dicyclic, and semidihedral groups.