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http://dx.doi.org/10.4134/CKMS.c210348

THE INDEPENDENCE AND INDEPENDENT DOMINATING NUMBERS OF THE TOTAL GRAPH OF A FINITE COMMUTATIVE RING  

Abughazaleh, Baha' (Department of Mathematics Isra University)
Abughneim, Omar AbedRabbu (Department of Mathematics The University of Jordan)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.4, 2022 , pp. 969-975 More about this Journal
Abstract
Let R be a finite commutative ring with nonzero unity and let Z(R) be the zero divisors of R. The total graph of R is the graph whose vertices are the elements of R and two distinct vertices x, y ∈ R are adjacent if x + y ∈ Z(R). The total graph of a ring R is denoted by 𝜏(R). The independence number of the graph 𝜏(R) was found in [11]. In this paper, we again find the independence number of 𝜏(R) but in a different way. Also, we find the independent dominating number of 𝜏(R). Finally, we examine when the graph 𝜏(R) is well-covered.
Keywords
Total graph of a commutative ring; zero divisors; independence number; independent dominating number; well-covered graphs;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 A. M. Dhorajia, Total graph of the ring ℤn × ℤm, Discrete Math. Algorithms Appl. 7 (2015), no. 1, 1550004, 9 pp. https://doi.org/10.1142/S1793830915500044   DOI
2 A. Mishra and K. Patra, Domination and independence parameters in the total graph of ℤn with respect to Nil ideal, IAENG Intern. J. Appl. Math. 50 (2020), no. 3, 707-712.
3 M. D. Plummer, Well-covered graphs: a survey, Quaestiones Math. 16 (1993), no. 3, 253-287.   DOI
4 A. Abbasi and S. Habibi, The total graph of a commutative ring with respect to proper ideals, J. Korean Math. Soc. 49 (2012), no. 1, 85-98. https://doi.org/10.4134/JKMS.2012.49.1.085   DOI
5 S. Akbari, D. Kiani, F. Mohammadi, and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra 213 (2009), no. 12, 2224-2228. https://doi.org/10.1016/j.jpaa.2009.03.013   DOI
6 N. Ananchuen, W. Ananchuen, and M. D. Plummer, Domination in graphs, in Structural analysis of complex networks, 73-104, Birkhauser/Springer, New York, 2011. https://doi.org/10.1007/978-0-8176-4789-6_4   DOI
7 D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706-2719. https://doi.org/10.1016/j.jalgebra.2008.06.02   DOI
8 D. F. Anderson and A. Badawi, The generalized total graph of a commutative ring, J. Algebra Appl. 12 (2013), no. 5, 1250212, 18 pp. https://doi.org/10.1142/S021949881250212X   DOI
9 W. Goddard and M. A. Henning, Independent domination in graphs: a survey and recent results, Discrete Math. 313 (2013), no. 7, 839-854. https://doi.org/10.1016/j.disc.2012.11.031   DOI
10 H. R. Maimani, C. Wickham, and S. Yassemi, Rings whose total graphs have genus at most one, Rocky Mountain J. Math. 42 (2012), no. 5, 1551-1560. https://doi.org/10.1216/RMJ-2012-42-5-1551   DOI
11 M. D. Plummer, Some covering concepts in graphs, J. Combinatorial Theory 8 (1970), 91-98.   DOI
12 K. Nazzal, Total graphs associated to a commutative ring, Palest. J. Math. 5 (2016), Special Issue, 108-126.
13 M. H. Shekarriz, M. H. Shirdareh Haghighi, and H. Sharif, On the total graph of a finite commutative ring, Comm. Algebra 40 (2012), no. 8, 2798-2807. https://doi.org/10.1080/00927872.2011.585680   DOI
14 W. Willis, Bounds for the independence number of a graph, Master Thesis in Virginia Commonwealth University, 2011.
15 D. F. Anderson and A. Badawi, On the total graph of a commutative ring without the zero element, J. Algebra Appl. 11 (2012), no. 4, 1250074, 18 pp. https://doi.org/10.1142/S0219498812500740   DOI