The total graph of a finite commutative ring

Ali RAMIN

Article Details

The total graph of a finite commutative ring

Authors:

Ali RAMIN

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2013 Vol.37 Issue 3, pp.391-397

Document Type:

Article

Publication Date:

2/2013

Document Language:

Turkish

Document ID (Org No.):

2000747591

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

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(G(R)), is the graph with all elements of  R as vertices, and two distinct vertices x,y \in R, are adjacent in  T(G(R)) if and only if x+y \in Z(R).  In this paper, some  properties of T(G(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(G(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.

 

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(G(R)), is the graph with all elements of  R as vertices, and two distinct vertices x,y \in R, are adjacent in  T(G(R)) if and only if x+y \in Z(R).  In this paper, some  properties of T(G(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(G(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.

Article Details

The total graph of a finite commutative ring

Authors:

Ali RAMIN

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2013 Vol.37 Issue 3, pp.391-397

Document Type:

Article

Publication Date:

2/2013

Document Language:

Turkish

Document ID (Org No.):

2000747591

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Commutative rings

total graph

regular elements

zero-divisors

Issues

Article Details

The total graph of a finite commutative ring

Authors: Ali RAMIN

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2013 Vol.37 Issue 3, pp.391-397

Document Type: Article

Publication Date: 2/2013

Document Language: Turkish

Document ID (Org No.): 2000747591

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Commutative rings

total graph

regular elements

zero-divisors

Issues

Authors:

Ali RAMIN

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2013 Vol.37 Issue 3, pp.391-397

Document Type:

Article

Publication Date:

2/2013

Document Language:

Turkish

Document ID (Org No.):

2000747591

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098