A graph associated to proper non-small subsemimodules of a semimodule

Let M be a unitary left R-semimodule where R is a commutative semiring with identity. The  small intersection graph G(M) of a semimodule M is an undirected simple graph with all non-  small proper subsemimodules of M as vertices and two distinct vertices N and L are adjacent if  and only if N ∩ L is not small in M. In this paper, we investigate the fundamental properties  of these graphs to relate the combinatorial properties of G(M) to the algebraic properties of the  R-semimodule M. Determine the diameter and the girth of G(M). Moreover, we study cut vertex,  clique number, domination number and independence number of the graph G(M). It is shown that  the independence number of small graph is equal to the number of its maximal subsemimodules.