Extended total graph associated to finite commutative rings
UDC 512.5 For a commutative ring $R$  with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ is a simple graph in which all elements of $R$ are vertices and any two distinct vertices $x$ an...
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| Date: | 2024 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7494 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 512.5
For a commutative ring $R$  with nonzero identity $1\neq 0$, let $Z(R)$ denote the set of zero divisors. The total graph of $R$ denoted by $T_{\Gamma}(R)$ is a simple graph in which all elements of $R$ are vertices and any two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this paper, we define an extension of the total graph denoted by $T(\Gamma^{e}(R))$ with vertex set as  $Z(R),$ and two distinct vertices $x$ and $y$ are adjacent if and only if  $x+y\in Z^*(R)$, where $ Z^{*}(R)$ is the set of nonzero zero divisors of $R$.  Our main aim is to characterize the finite commutative rings  whose $T(\Gamma^{e}(R))$ has clique numbers  $1,2,$ and $3$.  In addition, we characterize finite commutative nonlocal rings $R$ for which  the corresponding graph $T(\Gamma^{e}(R))$ has the clique number $4.$ |
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| DOI: | 10.3842/umzh.v76i5.7494 |