On the Genus of the Total Graph of a Commutative Ring

Abstract

Let R be a commutative ring and Z(R) be its set of all zero-divisors. The total graph of R, denoted by T_Γ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Maimani et al. [13] determined all isomorphism classes of finite commutative rings whose total graph has genus at most one. In this article, after enumerating certain lower and upper bounds for genus of the total graph of a commutative ring, we characterize all isomorphism classes of finite commutative rings whose total graph has genus two.

Key Words:

• Annihilator ideal
• Commutative ring
• Genus
• Total graph

2000 Mathematics Subject Classification:

• 05C75
• 05C25
• 13A15
• 13M05

ACKNOWLEDGMENT

The authors express their sincere thanks to the referee for many suggestions which improved the presentation of the article considerably. The work reported here is supported by the UGC Major Research Project F. No. 37-267/2009(SR) awarded to the first author by the University Grants Commission, Government of India. Also the work is supported by the INSPIRE programme (IF 110072) of Department of Science and Technology, Government of India for the second author.

Notes

Communicated by I. Swanson.