Characterization of rings with genus two prime ideal sum graphs

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus $2$.Comment: 13 figures, Asian-European Journal of Mathematics, Accepte

On the connection between the topological graph property and the theory of commutative rings

We try to establish some connections between commutative ring theory and topological graph theory, by study of the genus of the intersection graph of ideals and classify all graphs of genus 1 and genus 2 that are intersection graphs of ideals of some commutative rings

Embedding of prime ideal sum graph of a commutative ring on surfaces

Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we classify non-local commutative rings $R$ such that $\text{PIS}(R)$ is of crosscap at most two. We prove that there does not exist a finite non-local commutative ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative rings of genus one prime ideal sum graphs. Moreover, we classify finite non-local commutative rings for which the prime ideal sum graph is split graph, threshold graph, cograph, cactus graph and unicyclic, respectively

Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring I, nonquasilocal case

The rings considered in this article are nonzero commutative with identity which are not fields. Let R be a ring. We denote the collection of all proper ideals of R by I(R) and the collection I(R)\{(0)} by I(R)*. Recall that the intersection graph of ideals of R, denoted by G(R), is an undirected graph whose vertex set is I(R)* and distinct vertices I, J are adjacent if and only if I ∩ J ≠ (0). In this article, we consider a subgraph of G(R), denoted by H(R), whose vertex set is I(R)* and distinct vertices I, J are adjacent in H(R) if and only if IJ ≠ (0). The purpose of this article is to characterize rings R with at least two maximal ideals such that H(R) is planar

Planarity of a spanning subgraph of the intersection graph of ideals of a commutative ring II, Quasilocal Case

The rings we consider in this article are commutative with identity 1 ≠ 0 and are not fields. Let R be a ring. We denote the collection of all proper ideals of R by I(R) and the collection I(R) \ {(0)} by I(R)*. Let H(R) be the graph associated with R whose vertex set is I(R)* and distinct vertices I, J are adjacent if and only if IJ ≠ (0). The aim of this article is to discuss the planarity of H(R) in the case when R is quasilocal

The Analysis of the rings and modules using associated graphs

Ova doktorska disertacija prouqava razliqite osobine komutativnih prstena i modula algebarsko kombinatornim metodama. Ako se graf na odgovarajui naqin pridrui prstenu R ili R-modulu M, onda ispitiva em egovih osobina dolazimo do korisnih informacija o R i M. U ovoj tezi odreen je radijus totalnog grafa komutativnog prstena R u sluqaju kada je taj graf povezan. Tipiqna raxirea kao xto su prsten polinoma, prsten formalnih redova, idealizacija R-modula M i prsten matrica Mn(R) takoe su ispitani. Ustanov ene su veze izmeu totalnog grafa polaznog prstena R i totalnih grafova ovih raxirea. Definisaem totalnog grafa modula dato je jedno uopxtee totalnog grafa komutativnog prstena. Ispitane su i dokazane egove razliqite osobine. Ustanov ene su veze sa totalnim grafom prstena kao i neke veze sa grafom delite a nule. U ci u bo eg razumevaa qistih prstena, uveden je qisti graf C¡(R) komutativnog prstena sa jedinicom R. Deta no su ispitane egove osobine. Da im istraivaem qistih grafova dobijeni su dodatni rezultati vezani za druge klase komutativnih prstena. Jedan od predmeta ove teze je i istraivae osobina odgovarajueg linijskog grafa L(T¡(R)) totalnog grafa T¡(R). Data je kompletna klasifikacija svih komutativnih prstena qiji su linijski grafovi totalnog grafa planarni ili toroidalni. Dokazano je da za ceo broj g ¸ 0 postoji samo konaqno mnogo komutativnih prstena takvih da je °(L(T¡(R))) = g. U ovoj tezi su takoe klasifikovani svi toroidalni grafovi koji su grafovi preseka ideala komutativnog prstena R. Dato je i jedno pobo xae postojeih rezultata o planarnosti ovih grafova...This dissertation examines various properties of commutative rings and modules using algebraic combinatorial methods. If the graph is properly associated to a ring R or to an R-module M, then examination of its properties gives useful information about the ring R or R-module M. This thesis discusses the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with. The total graph of a module, a generalization of the total graph of a ring is presented. Various properties are proved and some relations to the total graph of a ring as well as to the zero-divisor graph are established. To gain a better understanding of clean rings and their relatives, the clean graph C¡(R) of a commutative ring with identity is introduced and its various proper- ties established. Further investigation of clean graphs leads to additional results concerning other classes of commutative rings. One of the topics of this thesis is the investigation of the properties of the cor- responding line graph L(T¡(R)) of the total graph T¡(R). The classi¯cation of all commutative rings whose line graphs of the total graph are planar or toroidal is given. It is shown that for every integer g ¸ 0 there are only ¯nitely many commutative rings such that °(L(T¡(R))) = g. Also, in this thesis all toroidal graphs which are intersection graphs of ideals of a commutative ring R are classi¯ed. An improvement over the previous results concerning the planarity of these graphs is presented..

Toroidality of Intersection Graphs of Ideals of Commutative Rings

Let R be a commutative ring with identity and G(R) its intersection graph. In this paper, all toroidal graphs that are intersection graphs are classified. An improvement over the previous results concerning the planarity of these graphs is also presented