The diameter of commuting graph of matrix ring of division rings

Abstract

The commuting graph of a non-commutative ring [Formula: see text] denoted by [Formula: see text] is a graph whose vertices are non-central elements of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text]. Let [Formula: see text] be a division ring, [Formula: see text] and [Formula: see text]. In this paper, we show that if [Formula: see text] or [Formula: see text] and the field [Formula: see text] has no extension of degree [Formula: see text] then [Formula: see text] is a connected graph whose diameter is less than five. Also we prove that the diameter of [Formula: see text] is four where [Formula: see text] is a quaternion algebra over a field [Formula: see text] which has no extension of degree [Formula: see text] and [Formula: see text].