On the non-nilpotent graphs of a group

Year 2017, Volume: 22 Issue: 22, 78 - 96, 11.07.2017

Deiborlang Nongsiang Promode Kumar Saikia

https://doi.org/10.24330/ieja.325927

Cited By: 5

https://izlik.org/JA38AP43SU

Abstract

 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.
 

Keywords

Non-nilpotent graph , finite group

References

• 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.