Commuting graphs and extremal centralizers - Dolinar, Gregor (Ars mathematica contemporanea, 2014)
We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra ?$M_n(\mathbb{F})$? over an arbitrary field ?$\mathbb{F}$?. It is known that if ?$\mathbb{F}$? is an algebraically closed field and ?$n \ge 3$?, then the diameter of the commuting graph of ?$M_n(\mathbb{F})$? is always equal to four. We construct a concrete example showing that if ?$\mathbb{F}$? is not algebraically closed, then the commuting graph of ?$M_n(\mathbb{F})$? can be connected with the diameter at least five.
Keywords: centralizatorji; centralizer; commuting graph; grafi komutativnosti; matrični kolobarji; matrix ring