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# g-Small Intersection Graph of a Module

## Authors

• Ahmed H. Alwan Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq. https://orcid.org/0000-0003-4367-4606

## Keywords:

Connectivity, Domination, Module, Small submodule, Small intersection graph

## Abstract

Let  be a commutative ring with identity, and  be a left -module. The g-small intersection graph of non-trivial submodules of , indicated by , is a simple undirected graph whose vertices are in one-to-one correspondence with all non-trivial submodules of  and two distinct vertices are adjacent if and only if the intersection of the corresponding submodules is a g-small submodule of . In this article, the interplay among the algebraic properties of , and the graph properties of  are studied. Properties of  such as connectedness, and completeness are considered. Besides, the girth and the diameter of  are determined, as well as presenting a formula to compute the clique and domination numbers of . The graph  is complete if,  is a generalized hollow module or  is a direct sum of two simple modules, is proved.

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