The inclusion subsemimodule graph of a semimodule

Let R be an abelian semiring with unity, U be an R-semimodule. The inclusion subsemimodule graph of U, indicated IS(U), is a graph with nodes that all non-trivial subsemimodules of U and two different nodes N and L are adjacent if and only if N ⊂ L or L ⊂ N. In this worke, proved that if U is subtractive semimodule then IS(U) is not connected if is a direct sum of two simple R-semimodules. Besides, it has been proved that IS(U) is a complete graph if and only if U is a uniserial semimodule. girth, diameter, chromatic and clique numbers of IS(U) have been studied. and only if U.