Yıl 2017, Cilt 22, Sayı 22, Sayfalar 78 - 96 2017-07-11

On the non-nilpotent graphs of a group

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

152 138

 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
  • A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298(2) (2006), 468-492.
  • A. Abdollahi and M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra, 38(12) (2010), 4390-4403.
  • A. Azad, M. A. Iranmanesh, C. E. Praeger and P. Spiga, Abelian coverings of finite general linear groups and an application to their non-commuting graphs, J. Algebraic Combin., 34(4) (2011), 638-710.
  • J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
  • M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Appl. Math., 157(4) (2009), 833-837.
  • A. K. Das and D. Nongsiang, On the genus of the nilpotent graphs of finite groups, Comm. Algebra, 43(12) (2015), 5282-5290.
  • The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.6.4, 2013 (http://www.gap-system.org).
  • B. Huppert and N. Blackburn, Finite Groups, III, Springer-Verlag, Berlin, 1982.
  • B. H. Neumann, A problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, 21(4) (1976), 467-472.
  • A. Yu. Ol'shanskii, Geometry of De ning Relations in Groups, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • L. Pyber, The number of pairwise noncommuting elements and the index of the centre in a nite group, J. London Math. Soc., 35(2) (1987), 287-295.
  • D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 2, Springer-Verlag, New York, 1972.
  • D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York-Berlin, 1982.
  • D. M. Rocke, p-Groups with abelian centralizers, Proc. London Math. Soc., 30(3) (1975), 55-75.
  • R. Schmidt, Zentralisatorverbande endlicher gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970), 97-131.
  • R. M. Solomon and A. J. Woldar, Simple groups are characterized by their non-commuting graphs, J. Group Theory, 16(6) (2013), 793-824.
  • V. P. Sunkov, Periodic group with almost regular involutions, Algebra i Logika, 7(1) (1968), 113-121.
  • D. B. West, Introduction to Graph Theory (Second Edition), PHI Learning Private Limited, New Delhi, 2009.
  • A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, 8, American Elsevier Publishing Co., Inc., New York, 1973.
  • C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra, 36(2) (2008), 325-345.
Konular Matematik ve İstatistik
Dergi Bölümü Makaleler
Yazarlar

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 <http://dergipark.gov.tr/ieja/issue/30344/325927>
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