We construct two practical algorithms for twisted conjugacy classes of polycyclic groups. The first algorithm determines whether elements a group are conjugate given endomorphisms, under the condition that their Reidemeister coincidence number is finite. second representatives endomorphisms if finite, or returns “false” this infinite. also discuss theoretical extension these to polycyclic-by-fi...

emph{The (undirected) power graph on the conjugacy classes} $mathcal{P_C}(G)$ of a group $G$ is a simple graph in which the vertices are the conjugacy classes of $G$ and two distinct vertices $C$ and $C'$ are adjacent in $mathcal{P_C}(G)$ if one is a subset of a power of the other. In this paper, we describe groups whose associated graphs are $k$-regular for $k=5,6$.