The Independence and Independent Dominating Numbers of The Total Graph of a Finite Commutative Ring
Commun. Korean Math. Soc.
Published online May 13, 2022
Baha' Abughazaleh and Omar Abughneim
Isra University; The University of Jordan
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.
MSC numbers : 13M99; 05C69
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