An associated graph to the k-subsemimodules of a semimodule

Let be a commutative semiring with identity and be an unitary - semimodule. The associated graph ( ) is defined to be the undirected simple graph whose vertices are non-trivial k-subsemimodules of and two distinct vertices are adjacent if and only if the corresponding k-subsemimodules of have nonzero intersection. We investigate the interplay between the semimodule-theoretic properties of and the graph-theoretic properties of ( ). We determine the diameter and the girth of ( ).

Keywords:

graph , semimodule , k-subsemimodule , connected