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## Some properties of the total graph and regular graph of a commutative ring

#### Manal Ghanem [1] , Khalida Nazzal [2]

##### 41 134

Let $R$ be a commutative ring with unity. The total graph of $R$, $T(\Gamma(R))$, is the simple graph with vertex set $R$ and two distinct vertices are adjacent if their sum is a zero-divisor in $R$. Let Reg$(\Gamma(R))$ and $Z(\Gamma(R))$ be the subgraphs of $T(\Gamma(R))$ induced by the set of all regular elements and the set of zero-divisors in $R$, respectively. We determine when each of the graphs $T(\Gamma(R))$, Reg$(\Gamma(R))$, and $Z(\Gamma(R))$ is locally connected, and when it is locally homogeneous. When each of Reg$(\Gamma(R))$ and
$Z(\Gamma(R))$ is regular and when it is Eulerian.
Total graph of a commutative ring, Regular graph of a commutative ring, Locally connected, Locally homogeneous, Regular graph, Eulerian graph
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Primary Language en Mathematics Mathematics Author: Manal Ghanem Author: Khalida Nazzal Publication Date: August 1, 2018
 Bibtex @research article { hujms453103, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {Hacettepe University}, year = {2018}, volume = {47}, pages = {835 - 843}, doi = {}, title = {Some properties of the total graph and regular graph of a commutative ring}, key = {cite}, author = {Ghanem, Manal and Nazzal, Khalida} } APA Ghanem, M , Nazzal, K . (2018). Some properties of the total graph and regular graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics, 47 (4), 835-843. Retrieved from http://dergipark.org.tr/hujms/issue/38872/453103 MLA Ghanem, M , Nazzal, K . "Some properties of the total graph and regular graph of a commutative ring". Hacettepe Journal of Mathematics and Statistics 47 (2018): 835-843 Chicago Ghanem, M , Nazzal, K . "Some properties of the total graph and regular graph of a commutative ring". Hacettepe Journal of Mathematics and Statistics 47 (2018): 835-843 RIS TY - JOUR T1 - Some properties of the total graph and regular graph of a commutative ring AU - Manal Ghanem , Khalida Nazzal Y1 - 2018 PY - 2018 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 835 EP - 843 VL - 47 IS - 4 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2017 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Some properties of the total graph and regular graph of a commutative ring %A Manal Ghanem , Khalida Nazzal %T Some properties of the total graph and regular graph of a commutative ring %D 2018 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 47 %N 4 %R %U ISNAD Ghanem, Manal , Nazzal, Khalida . "Some properties of the total graph and regular graph of a commutative ring". Hacettepe Journal of Mathematics and Statistics 47 / 4 (August 2018): 835-843. AMA Ghanem M , Nazzal K . Some properties of the total graph and regular graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics. 2018; 47(4): 835-843. Vancouver Ghanem M , Nazzal K . Some properties of the total graph and regular graph of a commutative ring. Hacettepe Journal of Mathematics and Statistics. 2018; 47(4): 843-835.