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Chromatic number and clique number of subgraphs of regular graph of matrix algebras

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2011.09.020
  3. Abstract:
  4. Let R be a ring and X R be a non-empty set. The regular graph of X, Γ(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Γ( GLn(F)) finite? In this paper, we show that if G is a soluble subgroup of GLn(F), then χ(Γ(G))<∞. Also, we show that for every field F, χ(Γ( Mn(F)))=χ(Γ( Mn(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Γ(), where denotes the subgroup generated by A∈ GLn(F)
  5. Keywords:
  6. Determinant ; Regular graph ; Chromatic number ; Clique number ; Maximum values ; Regular graphs ; Subgraphs ; Zero divisors ; Manganese ; Graph theory
  7. Source: Linear Algebra and Its Applications ; Volume 436, Issue 7 , 2012 , Pages 2419-2424 ; 00243795 (ISSN)
  8. URL: http://www.sciencedirect.com/science/article/pii/S0024379511006537