Super commuting graphs of finite groups and their Zagreb indices

Abstract

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.