关键词： Von Neumann正则环, 零因子图, 连通性, 源点, 收点

Abstract: In this paper we investigate the zero-divisor graph of von Neumann regular rings，and we focus our main attention on the property of vertices and the connectedness of the zero-divisor graph Γ(R).Let R be a von Neumann regular ring with unit 1，we show that  Γ(R) is connected if and only if R is direct finite.In addition，if R is a regular ring without unity elements，then  Γ(R) is connected if and only if R has no proper one-sided identity elements.For the zero-divisor graph  Γ(R) of a regular ring R with |R|≥ 5，we have that Sour(R)={a∈R|a is right invertible but  not left invertible} and Sink (R)={a∈R|a is left invertible but not right invertible}.

Key words: Von Neumann regular rings, zero-divisor graphs, connectedness, sources, sinks