Perfect Nilpotent Graphs

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DOI: 10.46793/KgJMat2104.521N


Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set ZN(R), and two vertices x and y are adjacent if and only if xy is nilpotent, where ZN(R) = {x Rxy 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 ΓN(R) is perfect. In addition, it is shown that for a ring R, if R is Artinian, then ωN(R)) = χN(R)) = |Nil(R)| + |Max(R)|.


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


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