The Complexity of Commuting Graphs and Related Topics

‎Let $G$ be a group and associate with $G$ a graph ${\mathcal C} (G)$ as follows‎: ‎the vertices are the elements of $G$ and two vertices $x‎, ‎y$ are joined by an edge if and only if $x$ and $y$ commute as elements of $G$‎. ‎This graph‎, ‎which is always connected‎, ‎is called commuting graph of $G$‎. ‎In this talk‎, ‎we will provide some formulas concerning the complexities of commuting graphs‎ ‎associated with certain finite groups‎. ‎We will also classify all nonabelian groups‎, ‎up to isomorphism‎, ‎which have an $n$-abelian partition for $n=2‎, ‎3$‎.