The total graph of a finite commutative ring

Abstract

Let R be a commutative ring with Z(R) , its set of zero-divisors and Reg(R) , its set of regular elements. Total graph of R, denoted by T(Γ(R)) , is the graph with all elements of R as vertices, and two distinct vertices x, y ε R, are adjacent in T(Γ(R)) if and only if x+y ε Z(R) . In this paper, some properties of T(Γ(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 T(Γ(R)) coincides with the minimum degree if and only if R is a finite commutative ring such that Z(R) is not an ideal in R. © TÜBİTAK.