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 U₆₍, V₈₍, M₂₌₍ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vukicevi\'c conjecture.