A note on commuting graphs for general linear groups

Year 2019, Volume: 48 Issue: 6 , 1635 - 1641 , 08.12.2019

Maryam Nasiri Sayyed Heidar Jafari

https://izlik.org/JA57BU89ES

Abstract

Let $G$ be a group and $X$ a subset of $G$. Then $\mathcal{C}(G, X)$ is a graph with vertex set $X$ in which two distinct elements $x$, $y\in X$ are joined by an edge if $xy=yx$. In this paper, we study the clique number, the domination number, the diameter, the planarity, the perfection and regularity of $\mathcal{C}(G, X)$ where $G=GL(n,q)$ and $X$ is the set of transvections.

Keywords

commuting graph , transvections , clique , domination , perfection , planarity

References

• [1] P.J. Cameron and Q. Mary, Automorphisms of graphs, London E14Ns, U.K. Draft, 2001.
• [2] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. of Math. (2), 164 (1), 51–229, 2006.
• [3] H. Poliatsek, Irreducible Groups Generated by Transvections over Finite Fields of Characteristic Two, J. Algebra, 39, 328–333, 1976.
• [4] J.J. Rotman, An Introduction to the Theory of groups, Springer-Verlag Newyork, Inc. 1995.