Commuting graphs and extremal centralizers

Abstract

We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra M n(F) over an arbitrary field F. It is known that if F is an algebraically closed field and n ≥ 3, then the diameter of the commuting graph of Mn(F) is always equal to four. We construct a concrete example showing that if F is not algebraically closed, then the commuting graph of Mn(F) can be connected with the diameter at least five. Copyright © 2014 DMFA Slovenije.