The essence of commutativity relation on a given Magma is captured in its commuting graph which, by definition, is a simple graph on noncentral elements as vertices and where two distinct vertices a,b form an edge if ab=ba. The commuting graph can be an important tool in investigation of nonabelian Magmas which lack the notion of a commutator (say, in semigroups) but can also be studied in nonabelian groups or algebras.

In the talk we aim to show a few examples of isometry transfer problems, i.e., that in certain class of (semi)groups and matrix algebras the commuting graphs are isomorphic if and only if the corresponding algebraic sets are isomorphic. We will also consider realizability questions, i.e., which graph can be obtained as a commuting graph of a semigroup or of a group.

Time permitting, we will further address property recognition for matrix algebras, i.e., how to determine from commuting graph if a matrix has, say, rank-one or is diagonalizable, and/or review basic ideas in classification of homomorphisms of certain commuting (and anti-commuting) graphs.