The annihilating graph of a ring

Abstract

Let A be a commutative ring with unity. The annihilating graph of A, denoted by {{mathbb {G}}}(A), 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 {rm Ann}(I){rm Ann}(J)=0. For every commutative ring A, we study the diameter and the girth of {mathbb {G}}(A). Also, we prove that if {mathbb {G}}(A) is a triangle-free graph, then {mathbb {G}}(A) is a bipartite graph. Among other results, we show that if {mathbb {G}}(A) is a tree, then {mathbb {G}}(A) 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 {mathbb {G}}({{mathbb {Z}}}_n) is a complete or a planar graph. Finally, we compute the domination number of {mathbb {G}}({mathbb {Z}}_n).