  Let $B$ be an equivalence relation defined on a finite group $G$. The $B$
super commuting graph on $G$ is a graph whose vertex set is $G$ and two
distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist
$g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is
the $B$-equivalence class of $g \in G$. Considering $B$ as the equality,
conjugacy and same order relations on $G$, in this article, we discuss the
graph structures of equality/conjugacy/order super commuting graphs of certain
well-known families of non-abelian groups viz. dihedral groups, dicyclic
groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n},
M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show
that they satisfy Hansen-Vuki{\v{c}}evi{\'c} conjecture.
