The total graph of a finite commutative ring

Abstract

Let R be a commutative ring with Z(R), its set of zero-divisors and \mbox{Reg}(R), its set of regular elements. Total graph of R, denoted by T(\Gamma(R)), is the graph with all elements of R as vertices, and two distinct vertices x,y \in R, are adjacent in T(\Gamma(R)) if and only if x+y \in Z(R). In this paper, some properties of T(\Gamma(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(\Gamma(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.