Super commuting graphs of finite groups and their Zagreb indices

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' ∈ [g] and h' ∈ [h] such that $g'$ commutes with $h'$, where $[g]$ is the B-equivalence class of g ∈ 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 U₆ₙ, V₈ₙ, M₂ₘₙ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{{c}}evi{\'c} conjecture.