Super Graphs on Groups, I

Abstract Let G be a finite group. A number of graphs with the vertex set have been studied, including power graph, enhanced and commuting graph. These form hierarchy under inclusion edge sets, it is useful to study them together. In addition, several authors considered modifying definition these by choosing natural equivalence relation on group such as equality, conjugacy, or equal orders, joining two elements if there are in their class that adjacent original this way, we enlarge into second dimension. Using three graph types relations mentioned gives nine graphs, which general only coincide; find conditions for some other pairs equal. often define interesting classes groups, EPPO 2-Engel Dedekind groups. We properties new hierarchy. particular, characterize groups complete, most cases, dominant vertices (those joined all others). Also, give results about universality, perfectness, clique number.