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On the compressed essential graph of a module over a commutative ring
(World Scientific Publ Co Pte Ltd, 2021) Payrovi, S. H.; Babaei, S.; Sengelen Sevim, E.
Let R be a commutative ring and M be an R-module. The compressed essential graph of M, denoted by EG(M) is a simple undirected graph associated to M whose vertices are classes of torsion elements of M and two distinct classes [m] and [m '] are adjacent if and only if Ann(R)(m) + Ann(R)(m ') is an essential ideal of R. In this paper, we study diameter and girth of EG(M) and we characterize all modules for which the compressed essential graph is connected. Moreover, it is proved that omega(EG(M)) = |Ass(R)(M)|, whenever R is Noetherian and M is a finitely generated multiplication module with r(Ann(R)(M)) = 0.
The extended graph of a module over a commutative ring
(World Scientific Publ Co Pte Ltd, 2023) Babaei, S.; Sengelen Sevim, E.
In this paper, we associate a simple graph to a module over a commutative ring and call it extended graph, that adjacent condition depends on the properties of annihilator submodules. We determine the structure of modules, when the associated graph is star or complete. Moreover, we look for relationships between the extended graph and the zero-divisor graph of modules.