### Perfect Nilpotent Graphs

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**Authors:**M. J. NIKMEHR AND A. AZADI

**DOI:**10.46793/KgJMat2104.521N

**Abstract:**

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by Γ

_{N}(R), is a graph with vertex set Z

_{N}(R)

^{∗}, and two vertices x and y are adjacent if and only if xy is nilpotent, where Z

_{N}(R) = {x ∈ R∣xy is nilpotent, for some y ∈ R

^{∗}}. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose Γ

_{N}(R) is perfect. In addition, it is shown that for a ring R, if R is Artinian, then ω(Γ

_{N}(R)) = χ(Γ

_{N}(R)) = |Nil(R)

^{∗}| + |Max(R)|.

**Keywords:**

Weakly perfect graph, perfect graph, chromatic number, clique number.

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