## Turkish Journal of Mathematics

#### Article Title

#### DOI

10.3906/mat-1101-70

#### 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.

#### Keywords

Commutative rings, total graph, regular elements, zero-divisors

#### First Page

391

#### Last Page

397

#### Recommended Citation

RAMIN, ALI
(2013)
"The total graph of a finite commutative ring,"
*Turkish Journal of Mathematics*: Vol. 37:
No.
3, Article 2.
https://doi.org/10.3906/mat-1101-70

Available at:
https://journals.tubitak.gov.tr/math/vol37/iss3/2