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The annihilating graph of a ring. (English) Zbl 1407.05125

Summary: 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 \(\mathrm{Ann}(I) \mathrm{Ann}(J)=0\). 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 \(\mathbb {G}(\mathbb {Z}_n)\).

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
05C40 Connectivity
13A05 Divisibility and factorizations in commutative rings
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References:

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