Some Properties of the Idempotent Graph of a Ring

Abstract

The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam(I(Mn(D))) = 4, for all natural number n ≥ 4 and diam(I(M3(D))) = 5, where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied.Mathematics Subject Classification. 05C15, 05C69, 17C27.