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

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 \( T(\Gamma(M)) \) 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 \( T(\Gamma(M)) \) and investigate the domination number and the bondage number of \( T(\Gamma(M)) \) and its induced subgraphs. Some domination parameters of \( T(\Gamma(M)) \) are also studied. It has been shown that \( T(\Gamma(M)) \) is excellent, domatically full, and well covered under certain conditions.