Rings whose total graphs have genus at most one

Details Rings whose total graphs have genus at most one Rings whose total graphs have genus at most one

2010-01-29

arxiv.org/abs/1001.5338v1

Mathematics

Commutative Algebra

7 pages. To appear in Rocky Mountain Journal of Mathematics

Scientific paper

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.

Affiliated with

Also associated with

No associations

Rating

Rings whose total graphs have genus at most one does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Rings whose total graphs have genus at most one, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rings whose total graphs have genus at most one will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-676332