Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R/H is a saturated multiplicatively closed subset of R. Anderson and Badawi [4] introduced the generalized total graph of R as an undirected simple graph GTH(R) with vertex set as R and any two distinct vertices x and y are adjacent if and only if x + y ϵ H. The main objective of this paper is to study the domination properties of the graph GTH(R). We determine the domination number of GTH(R) and its induced subgraphs GTH(H) and GTH(R/H). We establish a relationship between the domination number of GTH(R) and the same of GTH(R/H). We also establish a relationship between diameter and domination number of GTH(R/H). In addition,we obtain the bondage number of GTH(R). Finally, a relationship between girth and bondage number of GTH(R/H) has been established.
Patwari, D., Saikia, H., & Goswami, J. (2022). Some results on domination in the generalized total graph of a commutative ring. Journal of Algebra and Related Topics, 10(1), 119-128. doi: 10.22124/jart.2021.19238.1265
MLA
D. Patwari; H. K. Saikia; J. Goswami. "Some results on domination in the generalized total graph of a commutative ring". Journal of Algebra and Related Topics, 10, 1, 2022, 119-128. doi: 10.22124/jart.2021.19238.1265
HARVARD
Patwari, D., Saikia, H., Goswami, J. (2022). 'Some results on domination in the generalized total graph of a commutative ring', Journal of Algebra and Related Topics, 10(1), pp. 119-128. doi: 10.22124/jart.2021.19238.1265
VANCOUVER
Patwari, D., Saikia, H., Goswami, J. Some results on domination in the generalized total graph of a commutative ring. Journal of Algebra and Related Topics, 2022; 10(1): 119-128. doi: 10.22124/jart.2021.19238.1265
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