On the essential annihilating-ideal graph of commutative rings

Let R be a commutative ring with identity, and $Z\left(R\right)$ be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by $W\Gamma \left(R\right)$ is an undirected (simple) graph with vertex set $Z{\left(R\right)}^{*}$, and two distinct vertices x and y are adjacent, if and only if there exist $r\in ann\left(x\right)$ and $s\in ann\left(y\right)$, such that $rs=0$. Importantly, it is worth noting that $W\Gamma \left(R\right)$ contains the zero-divisor graph $\Gamma \left(R\right)$ as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring ${\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s$, denoted by $\mathrm{W}\Gamma ({\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s)$, where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of $\mathrm{W}\Gamma ({\mathbb{Z}}_p\times {\mathbb{Z}}_t\times {\mathbb{Z}}_s)$.

Let be a commutative ring with unity and let set of all zero divisors of be denoted by . An ideal of the ring is said to be essential if it has a nonzero intersection with every nonzero ideal of . It is denoted by . The generalized zero-divisor graph denoted by is an undirected graph with vertex set (set of all nonzero zero-divisors of ) and two distinct vertices and are adjacent if and only if . In this paper, first we characterized all the finite commutative rings for which is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings for which is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of .

Let u be a standard operator algebra on an infinite dimensional complex Hilbert space H containing identity operator I, which is closed under the adjoint operation. Suppose that delta : u -> u is the nonlinear mixed Jordan triple *-derivation. Then delta is an additive *-derivation.

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W S . The coannihilator graph of S , denoted by A G ′ S , is an undirected graph with vertex set W S ∗ (set of all nonzero nonunit elements of S ), and α ∼ β is an edge of A G ′ S ⇔ α ∉ α β S or β ∉ α β S , where δ S denotes the principal ideal generated by δ ∈ S . In this study, we first classify finite ring S , for which A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring S , for which A G ′ S is toroidal or projective.