Small Intersection Graph of Subsemimodules of a Semimodule

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.