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Domination number of total graphs

In: Mathematica Slovaca, vol. 66, no. 6
Abbas Shariatinia - Siamak Yassemi - Hamid Reza Maimani

Details:

Year, pages: 2016, 1527 - 1535
Keywords:
total graph, domination number, idealization (Nagata extension).
About article:
Let $R$ be a commutative ring with $Z(R)$ the set of zero-divisors and $U(R)$ the set of unit elements of $R$. The total graph of $R$, denoted by $T(Γ(R))$, is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x,y \in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y \in Z(R)$. We study the domination number of $T(Γ(R))$. It is shown that if $R=Z(R)\cup U(R)$, then the domination number of $T(Γ(R))$ is finite provided $R$ has a maximal ideal of finite index. Moreover, if $R=\prodi=1n Fi$, where $Fi$ is a field for each $1≤ i≤ n$ and $t=|F1|≤ |F2|≤ …≤|Fn|$, then the domination number of $T(Γ(R))$ is equal to $t-1$ provided $t=|Fi|$ for every $1≤ i≤ n$, and is equal to $t$ otherwise. Finally, for an $R$-module $M$ it is shown that the total domination number of the total graph of the idealization (Nagata extension) $R(+)M$ is equal to the domination number of the total graph of $R$ provided $M$ is a torsion free $R$-module or $R=Z(R)\cup U(R)$.
How to cite:
ISO 690:
Shariatinia, A., Yassemi, S., Maimani, H. 2016. Domination number of total graphs. In Mathematica Slovaca, vol. 66, no.6, pp. 1527-1535. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0241

APA:
Shariatinia, A., Yassemi, S., Maimani, H. (2016). Domination number of total graphs. Mathematica Slovaca, 66(6), 1527-1535. 0139-9918. DOI: https://doi.org/10.1515/ms-2016-0241
About edition:
Published: 1. 12. 2016