Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph Γ G (called noncommuting graph of G) with G as follows: Take G\Z(G) as the vertices of Γ G and join two distinct vertices x and y, whenever xy = yx. We want to explore how the graph theoretical properties of Γ G can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that Γ G ∼ = Γ H , then |G| = |H |. Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that Γ G ∼ = Γ H and |G| = |H |, then H is nilpotent.
Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)
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