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## On the non-nilpotent graphs of a group

#### Deiborlang Nongsiang [1] , Promode Kumar Saikia [2]

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Let $G$ be a group and $nil(G)=\{x \in G \mid \langle x,y \rangle \text{ is nilpotent for all }\\ y \in G\}$.
Associate a graph $\mathfrak{R}_G$ (called the non-nilpotent graph of $G$) with $G$ as follows: Take $G \setminus nil(G)$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of $\mathfrak{R}_G$. We conjecture that the domination number of the non-nilpotent graph of every finite non-abelian simple group is 2. We also conjecture that if $G$ and $H$ are two non-nilpotent finite groups such that $\mathfrak{R}_G\cong \mathfrak{R}_H$, then $|G| = |H|$. Among other results, we show that the non-nilpotent graph of $D_{10}$ is double-toroidal.

Non-nilpotent graph, finite group
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Konular Matematik Makaleler Yazar: Deiborlang Nongsiang Yazar: Promode Kumar Saikia
 Bibtex @araştırma makalesi { ieja325927, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {Prof. Dr. Abdullah HARMANCI}, year = {2017}, volume = {22}, pages = {78 - 96}, doi = {10.24330/ieja.325927}, title = {On the non-nilpotent graphs of a group}, key = {cite}, author = {Saikia, Promode Kumar and Nongsiang, Deiborlang} } APA Nongsiang, D , Saikia, P . (2017). On the non-nilpotent graphs of a group. International Electronic Journal of Algebra, 22 (22), 78-96. DOI: 10.24330/ieja.325927 MLA Nongsiang, D , Saikia, P . "On the non-nilpotent graphs of a group". International Electronic Journal of Algebra 22 (2017): 78-96 Chicago Nongsiang, D , Saikia, P . "On the non-nilpotent graphs of a group". International Electronic Journal of Algebra 22 (2017): 78-96 RIS TY - JOUR T1 - On the non-nilpotent graphs of a group AU - Deiborlang Nongsiang , Promode Kumar Saikia Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.325927 DO - 10.24330/ieja.325927 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 78 EP - 96 VL - 22 IS - 22 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.325927 UR - http://dx.doi.org/10.24330/ieja.325927 Y2 - 2018 ER - EndNote %0 International Electronic Journal of Algebra On the non-nilpotent graphs of a group %A Deiborlang Nongsiang , Promode Kumar Saikia %T On the non-nilpotent graphs of a group %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 22 %N 22 %R doi: 10.24330/ieja.325927 %U 10.24330/ieja.325927 ISNAD Nongsiang, Deiborlang , Saikia, Promode Kumar . "On the non-nilpotent graphs of a group". International Electronic Journal of Algebra 22 / 22 (Temmuz 2017): 78-96. http://dx.doi.org/10.24330/ieja.325927