Geodesic
Perfect Nilpotent Graphs
Kragujevac Journal of Mathematics, Tome 45 (2021) no. 4, p. 521 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

Let $R$ be a commutative ring with identity. The nilpotent graph of $R$, denoted by $\Gamma_N(R)$, 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 $Z_N(R)= \{x \in R \mid xy$ is nilpotent, for some $y \in 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 $\Gamma_N(R)$ is perfect. In addition, it is shown that for a ring $R$, if $R$ is Artinian, then $\omega(\Gamma_N(R))=\chi(\Gamma_N(R))=|{\rm Nil}(R)^*|+|{\rm Max}(R)|$.
Classification : 05c15, 05C17, 05C15, 05C25
Keywords: weakly perfect graph, perfect graph, chromatic number, clique number 
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     author = {M. J. Nikmehr and A. Azadi},
     title = {Perfect {Nilpotent} {Graphs}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {521 },
     publisher = {mathdoc},
     volume = {45},
     number = {4},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2021_45_4_a1/}
}
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M. J. Nikmehr; A. Azadi. Perfect Nilpotent Graphs. Kragujevac Journal of Mathematics, Tome 45 (2021) no. 4, p. 521 . http://geodesic.mathdoc.fr/item/KJM_2021_45_4_a1/