Some results on the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let $R$ be a ring such that $R$ admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of $R$, denoted by $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$

Harmonic Mean Cordial labeling of some graphs

All the graphs considered in this article are simple and undirected. Let G = (V(G), E(G)) be a simple undirected Graph. A function f : V (G) → {1, 2} is called Harmonic Mean Cordial if the induced function f*: E(G) → {1, 2} defined by f* (uv) = [2f(u)f(v)/f(u)+f(v)] satisfies the condition |vf (i) − vf (j)| ≤ 1 and |ef (i) − ef (j)| ≤ 1 for any i, j ∈ {1, 2}, where vf (x) and ef (x) denotes the number of vertices and number of edges with label x respectively and bxc denotes the greatest integer less than or equals to x. A Graph G is called Harmonic Mean Cordial graph if it admits Harmonic Mean Cordial labeling. In this article, we have provided some graphs which are not Harmonic Mean Cordial and also we have provided some graphs which are Harmonic Mean Cordial.Publisher's Versio

Annihilating-ideal graphs with independence number at most four

Let R be a commutative non-domain ring with identity and let ${ \mathbb A }(R)^{*}$ denote the set of all nonzero annihilating ideals of R. Recall that the annihilating-ideal graph of R, denoted by ${ \mathbb A }{ \mathbb G }(R)$, is an undirected simple graph whose vertex set is ${ \mathbb A }(R)^{*}$ and distinct vertices I, J are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article was to classify commutative rings R such that the independence number of ${ \mathbb A }{ \mathbb G }(R)$ is less than or equal to four