@article { author = {Visweswaran, S. and Parejiya, J.}, title = {A Note on a graph associated to a commutative ring}, journal = {Journal of Algebra and Related Topics}, volume = {5}, number = {1}, pages = {61-82}, year = {2017}, publisher = {University of Guilan}, issn = {2345-3931}, eissn = {2382-9877}, doi = {10.22124/jart.2017.2399}, abstract = {The rings considered in this article are commutative with identity. This article is motivated by the work on comaximal graphs of rings.  In this article, with any ring $R$, we associate an undirected graph denoted by $G(R)$, whose vertex set is the set of all elements of $R$ and distinct vertices $x,y$ are joined by an edge in $G(R)$ if and only if $Rx\cap Ry = Rxy$.  In Section 2 of this article, we classify rings $R$ such that $G(R)$ is complete and we also consider the problem of determining rings $R$ such that $\chi(G(R)) = \omega(G(R))< \infty$. In Section 3 of this article, we classify rings $R$ such that $G(R)$ is planar.}, keywords = {Comaximal graph of a ring,complete graph,chromatic number,clique number,planar graph}, url = {https://jart.guilan.ac.ir/article_2399.html}, eprint = {https://jart.guilan.ac.ir/article_2399_1591422012d4680950f3bb1760b533a6.pdf} }