On commuting graphs of semisimple rings. (English) Zbl 1063.05087
Authors’ abstract: Let \(R\) be a non-commutative ring. The commuting graph of \(R\), denoted by \(\Gamma(R)\), is a graph with vertex set \(R\setminus Z(R)\), and two distinct vertices \(a\) and \(b\) are adjacent if \(ab=ba\). In this paper we investigate some properties of \(\Gamma(R)\), whenever \(R\) is a finite semisimple ring. For any finite field \(F\), we obtain minimum degree, maximum degree and clique number of \(\Gamma(M_n(F))\). Also it is shown that for any two finite semisimple rings \(R\) and \(S\), if \(\Gamma(R)\simeq\Gamma(S)\), then there are commutative semisimple rings \(R_1\) and \(S_1\) and a semisimple ring \(T\) such that \(R\simeq T\times R_1\), \(S\simeq T\times S_1\) and \(|R_1|=|S_1|\).
Reviewer: Dragoš Cvetković (Beograd)
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 16P10 | Finite rings and finite-dimensional associative algebras |
| 15A27 | Commutativity of matrices |
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