On the Unit Graph of a Noncommutative Ring

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Let R be a ring (not necessary commutative) with non-zero identity. The unit graph of R, denoted by G(R), is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a+b is a unit element of R. It was proved that if R is a commutative ring and is a maximal ideal of R such that |R/|=2, then G(R) is a complete bipartite graph if and only if (R,) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessary commutative), then G(R) is a complete r-partite graph if and only if (R,) is a local ring and r=|R/m|=2ⁿ, for some n ∈ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring.