Volume 54, Issue 4
Domination in Generalized Cayley Graph of Commutative Rings

J. Math. Study, 54 (2021), pp. 427-434.

Published online: 2021-06

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• Abstract

Let $R$ be a commutative ring with identity and $n$ be a natural number. The generalized Cayley graph of $R$, denoted by $Γ^n_R$, is the graph whose vertex set is $R^n$\{0} and two distinct vertices $X$ and $Y$ are adjacent if and only if there exists an $n×n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are non-zero such that $AX^T=Y^T$ or $AY^T=X^T$, where for a matrix $B$, $B^T$ is the matrix transpose of $B$. In this paper, we give some basic properties of $Γ^n_R$ and we determine the domination parameters of $Γ^n_R$.

• Keywords

Ring, Cayley graph, generalized Cayley graph, domination number.

13A99, 05C69

selva_158@yahoo.co.in (K. Selvakumar)

subashini.m05@gmail.com (M. Subajini)

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@Article{JMS-54-427, author = {K. and Selvakumar and selva_158@yahoo.co.in and 16787 and Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India and K. Selvakumar and M. and Subajini and subashini.m05@gmail.com and 16788 and Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India and M. Subajini and S. and Pirzada and pirzadasd@kashmiruniversity.ac.in and 16789 and Department of Mathematics, University of Kashmir, Hazratbal, Srinagar, Kashmir, India and S. Pirzada}, title = {Domination in Generalized Cayley Graph of Commutative Rings}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {4}, pages = {427--434}, abstract = {

Let $R$ be a commutative ring with identity and $n$ be a natural number. The generalized Cayley graph of $R$, denoted by $Γ^n_R$, is the graph whose vertex set is $R^n$\{0} and two distinct vertices $X$ and $Y$ are adjacent if and only if there exists an $n×n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are non-zero such that $AX^T=Y^T$ or $AY^T=X^T$, where for a matrix $B$, $B^T$ is the matrix transpose of $B$. In this paper, we give some basic properties of $Γ^n_R$ and we determine the domination parameters of $Γ^n_R$.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n4.21.07}, url = {http://global-sci.org/intro/article_detail/jms/19296.html} }
TY - JOUR T1 - Domination in Generalized Cayley Graph of Commutative Rings AU - Selvakumar , K. AU - Subajini , M. AU - Pirzada , S. JO - Journal of Mathematical Study VL - 4 SP - 427 EP - 434 PY - 2021 DA - 2021/06 SN - 54 DO - http://doi.org/10.4208/jms.v54n4.21.07 UR - https://global-sci.org/intro/article_detail/jms/19296.html KW - Ring, Cayley graph, generalized Cayley graph, domination number. AB -

Let $R$ be a commutative ring with identity and $n$ be a natural number. The generalized Cayley graph of $R$, denoted by $Γ^n_R$, is the graph whose vertex set is $R^n$\{0} and two distinct vertices $X$ and $Y$ are adjacent if and only if there exists an $n×n$ lower triangular matrix $A$ over $R$ whose entries on the main diagonal are non-zero such that $AX^T=Y^T$ or $AY^T=X^T$, where for a matrix $B$, $B^T$ is the matrix transpose of $B$. In this paper, we give some basic properties of $Γ^n_R$ and we determine the domination parameters of $Γ^n_R$.

K. Selvakumar, M. Subajini & S. Pirzada. (2021). Domination in Generalized Cayley Graph of Commutative Rings. Journal of Mathematical Study. 54 (4). 427-434. doi:10.4208/jms.v54n4.21.07
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