On annihilator graph of a finite commutative ring

Document Type : Research Paper

Authors

1 North eastern Hill University

2 North Eastern Hill University

Abstract

‎The annihilator graph $AG(R)$ of a commutative ring $R$ is a simple undirected graph with the vertex set $Z(R)^*$ and two distinct vertices are adjacent if and only if $ann(x) \cup ann(y)$ $ \neq $ $ann(xy)$‎. ‎In this paper we give the sufficient condition for a graph $AG(R)$ to be complete‎. ‎We characterize rings for which $AG(R)$ is a regular graph‎, ‎we show that $\gamma (AG(R))\in \{1,2\}$ and we also characterize the rings for which $AG(R)$ has a cut vertex‎. ‎Finally we find the clique number of a finite reduced ring and characterize the rings for which $AG(R)$ is a planar graph‎.

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References

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History

  • Receive Date: 02 July 2015
  • Revise Date: 16 February 2016
  • Accept Date: 31 July 2016
  • Published Online: 01 March 2017

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