Document Type : Research Paper

Authors

• Ahmed H. Alwan
• Zahraa A. Nema

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

Abstract

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 $\mathrm{\Gamma }(U)$, 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 $N+L\ne U$. We study these graphs to relate the combinatorial properties of $\mathrm{\Gamma }(U)$ to the algebraic properties of the $S$-semimodule $U$. We study the connectedness of $\mathrm{\Gamma }(U)$. We investigate some properties of $\mathrm{\Gamma }(U)$ for instance, we find the domination number and clique number of $\mathrm{\Gamma }(U)$. Also, we study cycles in $\mathrm{\Gamma }(U)$.

Keywords

• Semimodule
• Co-intersection graph
• Connectivity
• Domination number
• Clique number