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 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 ∈ 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 ΓN(R) is perfect. In addition, it is shown that for a ring R, if R is Artinian, then ω(ΓN(R)) = χ(ΓN(R)) = |Nil(R)∗| + |Max(R)|.
Keywords:
Weakly perfect graph, perfect graph, chromatic number, clique number.
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