Some domination properties of the total graph of a module with respect to singular submodule

Abstract

Let $R$ be a commutative ring with unity and $M$ be an $R$-module. The total graph of $M$ with respect to the singular submodule $Z(M)$ of $M$ is an undirected graph $T(\mathrm{\Gamma }(M))$ with vertex set as $M$ and any two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(M)$. In this paper, the author attempts to study the domination in the graph $T(\mathrm{\Gamma }(M))$ and investigate the domination number and the bondage number of $T(\mathrm{\Gamma }(M))$ and its induced subgraphs. Some domination parameters of $T(\mathrm{\Gamma }(M))$ are also studied. It has been shown that $T(\mathrm{\Gamma }(M))$ is excellent, domatically full, and well covered under certain conditions.