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

Document Type : Research Paper


Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq


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.