On the co-intersection graph of subsemimodules of a semimodule

Let S be a semiring with identity and U be a unitary left S-semimodule. The co-intersection graph of an S-semimodule U, denoted by Γ(U), is defined to be the undirected simple graph whose vertices are in one-to-one correspondence with all non-trivial subsemimodules of U, and there is an edge between two distinct vertices N and L if and only if N+L≠U. We study these graphs to relate the combinatorial properties of Γ(U) to the algebraic properties of the S-semimodule U. We study the connectedness of Γ(U). We investigate some properties of Γ(U) for instance, we find the domination number and clique number of Γ(U). Also, we study cycles in Γ(U).