  In an earlier paper, the authors considered three types of graphs, and three
equivalence relations, defined on a group, viz.\ the power graph, enhanced
power graph, and commuting graph, and the relations of equality, conjugacy, and
same order; for each choice of a graph type A and an equivalence relation B,
there is a graph, the \emph{B superA graph} defined on $G$. The resulting nine
graphs (of which eight were shown to be in general distinct) form a
two-dimensional hierarchy. In the present paper, we consider these graphs
further. We prove universality properties for the conjugacy supergraphs of
various types, adding the nilpotent, solvable and enhanced power graphs to the
commuting graphs considered in the rest of the paper, and also examine their
relation to the invariable generating graph of the group. We also show that
supergraphs can be expressed as graph compositions, in the sense of Schwenk,
and use this representation to calculate their Wiener index. We illustrate
these by computing Wiener index of equality supercommuting and conjugacy
supercommuting graphs for dihedral and quaternion groups.
