On Oriented Diameter of Power Graphs (2409.02457v4)

Abstract: In this paper, we study the oriented diameter of power graphs of groups. We show that a $2$-edge connected power graph of a finite group has oriented diameter at most $4$. We prove that the power graph of the cyclic group of order $n$ has oriented diameter $2$ for all $n\neq 1,2,4,6$. For non-cyclic finite nilpotent groups, we show that the oriented diameter of corresponding power graphs is at least $3$. Moreover, we provide necessary and sufficient conditions for the oriented diameter of $2$-edge connected power graphs of finite non-cyclic nilpotent groups to be either $3$ or $4$. This, in turn, gives an algorithm for computing the oriented diameter of the power graph of a given nilpotent group that runs in time polynomial in the size of the group.