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Paper Information

Journal:   MATHEMATICAL SCIENCES   2018 , Volume 12 , Number 1; Page(s) 1 To 6.

The annihilating graph of a ring

Author(s):  SHAFIEi Z., MAGHASEDI M.*, HEYDARI F., Khojasteh S.
* Department of Mathematics, Karaj Azad University, Karaj, Iran
Let A be a commutative ring with unity. The annihilating graph of A, denoted by Gð AÞ , is a graph whose vertices are all non-trivial ideals of A and two distinct vertices I and J are adjacent if and only if Annð IÞ Annð JÞ ¼ 0. For every commutative ring A, we study the diameter and the girth of Gð AÞ . Also, we prove that if Gð AÞ is a triangle-free graph, then Gð AÞ is a bipartite graph. Among other results, we show that if Gð AÞ is a tree, then Gð AÞ is a star or a double star graph. Moreover, we prove that the annihilating graph of a commutative ring cannot be a cycle. Let n be a positive integer number. We classify all integer numbers n for which Gð ZnÞ is a complete or a planar graph. Finally, we compute the domination number of Gð ZnÞ .
Keyword(s): Annihilating graph ,Diameter ,Girth ,Planarity
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