@article{Nikandish_Nikmehr_Hosseini_2019, title={On the planarity and perfectness of annihilator ideal graphs}, volume={50}, url={https://journals.math.tku.edu.tw/index.php/TKJM/article/view/2707}, DOI={10.5556/j.tkjm.50.2019.2707}, abstractNote={<p>Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{Ann } (R) $, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if<br />$ I \cap \mathrm{Ann} _{R} (J)
eq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I)
eq \lbrace 0\rbrace $.<br />In this paper, all rings with planar annihilator ideal graphs are classified.<br />Furthermore, we show that all annihilator ideal graphs are perfect. Among other results, it is proved that if $\Gamma _{\mathrm{Ann } (R) $ is a tree, then $\Gamma _{\mathrm{Ann } (R) $ is star.</p>}, number={4}, journal={Tamkang Journal of Mathematics}, author={Nikandish, Reza and Nikmehr, M. J and Hosseini, S. M}, year={2019}, month={Dec.}, pages={361-369} }