Abstract We determine the metric dimension of annihilating-ideal graph a local finite commutative principal ring and with two maximal ideals. also find bounds for an arbitrary ring.

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator $R$ by $Ann(I)$. An is said to be exact annihilating if there exists non-zero $J$ such that $Ann(I) = J$ and $Ann(J) I$. set all ideals $\mathbb{EA}(R)$ $\mathbb{EA}(R)\backslash \{(0)\}$ $\mathbb{EA}(R)^{*}$. Let $\mathbb{EA}(R)^{*}\neq \emptyset$. With [Exact Annihilat...

Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undi...

Red and blue particles are placed in equal proportion throughout either the complete or star graph iteratively sampled to take simple random walk steps. Mutual annihilation occurs when with different colors meet. We compare time it takes extinguish every particle analogous (simple analyze) one-type setting. Additionally, we study effect of asymmetric speeds.

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

Let L be a lattice. The annihilating-ideal graph of is simple whose vertex set the all nontrivial ideals and two distinct vertices I J are adjacent if only I∧J=0. In this paper, crosscap graphs lattices with at most four atoms characterized. These characterizations provide classes multipartite graphs, which embedded in Klein bottle.