The necessity of commuting equality in groups

There are very few restrictions placed on the commuting graphs of semigroups. In particular, semigroups don't necessarily have elements that are commutativity equal. Groups on the other hand, tend to have have lots of elements that are commuting equal. Indeed, every single finite group of order greater then one has distinct elements that have the same centralizers.

Lemma. every group with max order two is commutative

Proof. We have that every element is equal to its own inverse. Therefore, $xy =(xy)^{-1} = y^{-1} x^{-1} = yx$. $\square$

Theorem. non-trivial finite groups have commuting equal elements

Proof. There are two possible cases (1) the group has max order two in which case by the previous lemma the group is totally commutative, which means that every element is commutativity equal to all the others or (2) the group has at least one element with order $n$ greater then or equal to three, in which case that element has totient $\varphi(n)$ iteration-equal elements. Iteration equal elements are commutativity-equal, thus the element which is not order two must have at least one element for which it is commutativity equal. $\square$

This proves commutativity equal elements exist in all groups from the smallest to the largest. To get that they not only always exist, but that they tend to be numerous consider that Cauchy's theorem implies that every group of order $n$ has a cyclic group of order $p$ for every prime number dividing $n$. Which means that the group must have a class of $\varphi(p)=p-1$ elements all of which are commutativity equal to each other.

For example, the symmetric group on three elements $S_3$ is the smallest non-commutative group. It has two commuting equal elements: the two elements of order three representing the two different cycles on three elements (1 2 3) and (1 3 2). On the other hand, the monster group has a class of at least seventy commutativity-equal elements as determined by its prime divisor of order 71. Not to mention the other commutativity-equal elements determined by all its other prime divisors. This demonstrates the importance of commuting equality in groups both larger and small.

Condensed commuting graphs:
Let $G$ be a graph, and $P$ its adjacency equivalence relation, then there is a natural graph homomorphism $f: G \to \frac{G}{P}$ where $\frac{G}{P}$ is the graph formed by condensing all adjacency equivalence classes down to single elements. The necessity of commutativity-equality suggests that we can apply this procedure to the commuting graphs of groups.

Star graphs:
When condensed, the commuting graphs of many small groups become star graphs. The center of the star graph is the center of the group condensed down to a single element. Commutative groups condense down to star graphs with a single element $S_1$. The symmetric group on three elements is a star graph on five elements $S_5$. The dihedral group on eight elements and the quaternion group both have $S_4$ as condensed commuting graphs, with the equivalent elements distributed equally in both of them. The dihedral group on ten elements has $S_7$. D12 and C3 : C4 have $S_5$ and A12 has $S_6$ as condensed commuting graphs. Things get more interesting when you consider certain larger groups.