Original Research Article

Article volume = 2022 and issue = 1

Pages: 15–22

Article publication Date: November 21, 2022

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# Small Intersection Graph of Subsemimodules of a Semimodule

#### Ahmed H. Alwan

Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq.

##### Abstract:

Let $R$ be a commutative semiring with identity, and $U$ be a unitary left $R$-semimodule. The small intersection graph of non-trivial subsemimodules of $U$, denoted by $\Gamma(U)$, is an undirected simple graph whose vertices are in one-to-one correspondence with all non-trivial subsemimodules of $U$ and two distinct vertices are adjacent if and only if the intersection of corresponding subsemimodules is a small subsemimodule of $U$. In this article, we investigate connections between the graph-theoretic properties of $\Gamma(U)$ and some algebraic properties of semimodules. We determine the diameter and the girth of $\Gamma(U)$. We obtain some results for connectivity and planarity of these graphs. Moreover, it is shown that the domination number of a small intersection graph of a semimodule is 1 , whenever $U$ is a subtractive semimodule and direct sum of two simple semimodules.

##### Keywords:

Semimodule, small subsemimodule, small intersection graph, Connectivity, domination number, planarity.

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##### Cite this article as:

- Ahmed H. Alwan, Small Intersection Graph of Subsemimodules of a Semimodule, Communications in Combinatorics, Cryptography & Computer Science, 2022(1), PP.15–22, 2022
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