On diameters of commuting graphs of matrix algebras over division rings

Abstract

Let [Formula: see text] be a real-closed field and [Formula: see text] a [Formula: see text]-division algebra. In this paper, we prove that if [Formula: see text] is algebraic over [Formula: see text] then the graph [Formula: see text] is connected and its diameter is at most [Formula: see text], for any [Formula: see text]. If in addition, the division ring [Formula: see text] is noncommutative, we also have the same results for [Formula: see text]. As a corollary, we show that the diameter of the commuting graph of the matrix algebra of degree [Formula: see text] over a generalized quaternion algebra [Formula: see text], where [Formula: see text] is a real-closed field, two elements [Formula: see text], is also at most [Formula: see text]. This fact is a strong improvement of the previous result by Akbari et al. asserting that this diameter is at most [Formula: see text] in case [Formula: see text] is the field [Formula: see text] of real numbers.