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 
@article{KJM_2021_45_4_a1,
     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/