Commuting Graphs of Matrix Algebras

Abstract

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))≃Γ(M _m (E)), then n = m and |F|=|E|.

Key Words:

• Commuting graph
• Matrix ring

2000 Mathematics Subject Classification:

• 05C50
• 15A27
• 16P10

ACKNOWLEDGMENTS

The first author is indebted to the Institute for Studies in Theoretical Physics and Mathematics (IPM) for support. The research of the first author was in part supported by a grant (No. 86050212) from IPM.

Notes

Communicated by M. Bresar.