The total graph of a finite commutative ring

Authors : Ali RAMIN

Pages : 391-397

Doi:10.3906/mat-1101-70

View : 14 | Download : 11

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let R be a commutative ring with Zinsert ignore into journalissuearticles values(R);, its set of zero-divisors and \mbox{Reg}insert ignore into journalissuearticles values(R);, its set of regular elements. Total graph of R, denoted by Tinsert ignore into journalissuearticles values(Ginsert ignore into journalissuearticles values(R););, is the graph with all elements of R as vertices, and two distinct vertices x,y \in R, are adjacent in Tinsert ignore into journalissuearticles values(Ginsert ignore into journalissuearticles values(R);); if and only if x+y \in Zinsert ignore into journalissuearticles values(R);. In this paper, some properties of Tinsert ignore into journalissuearticles values(Ginsert ignore into journalissuearticles values(R);); have been investigated, where R is a finite commutative ring and a new upper bound for vertex-connectivity has been obtained in this case. Also, we have proved that the edge-connectivity of Tinsert ignore into journalissuearticles values(Ginsert ignore into journalissuearticles values(R);); coincides with the minimum degree if and only if R is a finite commutative ring such that Zinsert ignore into journalissuearticles values(R); is not an ideal in R.
Keywords : Commutative rings, total graph, regular elements, zero divisors