International Journal of Nonlinear Analysis and Applications

On the co-intersection graph of subsemimodules of a semimodule

Document Type : Research Paper

Authors

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

Keywords

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International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 2763-2770
  • Receive Date: 06 April 2022
  • Revise Date: 15 May 2022
  • Accept Date: 04 June 2022