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The regular graph of a noncommutative ring

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1017/S0004972712001177
  3. Abstract:
  4. Let R be a ring and Z(R) be the set of all zero-divisors of R. The total graph of R, denoted by (TΓ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y in R are adjacent if and only if x + y Z(R). Let the regular graph of R, Reg (Γ(R)), be the induced subgraph of T(Γ (R)) on the regular elements of R. In 2008, Anderson and Badawi proved that the girth of the total graph and the regular graph of a commutative ring are contained in the set { 3, 4,} . In this paper, we extend this result to an arbitrary ring (not necessarily commutative). We also prove that if R is a reduced left Noetherian ring and 2 Z(R), then the chromatic number and the clique number of Reg ((R)) are the same and they are 2r , where r is the number of minimal prime ideals of R. Among other results, we show that if R is a semiprime left Noetherian ring and Reg (R) is finite, then R is finite
  5. Keywords:
  6. Chromatic number ; Girth ; Regular graph ; Total graph
  7. Source: Bulletin of the Australian Mathematical Society ; Vol. 89, issue. 1 , February , 2014 , pp. 132-140 ; ISSN: 00049727
  8. URL: http://journal.austms.org.au/ojs/index.php/Bulletin/article/view/6470