Distances on the Commuting Graph of the Ring of Real Martices

The vertices of the commuting graph of a semigroup $S$ are the noncentral elements of this semigroup, and its edges join all pairs of elements $g$, $h$ that satisfy the relation $gh=hg$. The paper presents a proof of the fact that the diameter of the commuting graph of the semigroup of real matrices of order $n\ge 3$ is equal to 4. A survey of results in that subject matter is presented, and several open problems are formulated.