On the edge metric dimension and Wiener index of the blow up of graphs

Let $G=(V,E)$ be a connected graph. The distance between an edge $e=xy$ and a vertex $v$ is defined as \T{d}(e,v)=\T{min}\{\T{d}(x,v),\T{d}(y,v)\}. A nonempty set $S \subseteq V(G)$ is an edge metric generator for $G$ if for any two distinct edges $e_1,e_2 \in E(G)$, there exists a vertex $s \in S$ such that \T{d}(e_1,s) \neq \T{d}(e_2,s). An edge metric generating set with the smallest number of elements is called an edge metric basis of $G$, and the number of elements in an edge metric basis is called the edge metric dimension of $G$ and it is denoted by \T{edim}(G). In this paper, we study the edge metric dimension of a blow up of a graph $G$, and also we study the edge metric dimension of the zero divisor graph of the ring of integers modulo $n$. Moreover, the Wiener index and the hyper-Wiener index of the blow up of certain graphs are computed

The annihilator ideal graph of a commutative ring

Let $R$ be a commutative ring with nonzero identity and $I$ be a proper ideal of $R$. The annihilator graph of $R$ with respect to $I$, which is denoted by $AG_{I}(R)$, is the undirected graph with vertex-set $V(AG_{I}(R)) = \lbrace x\in R \setminus I : xy \in I\$ for some$\ y \notin I \rbrace$ and two distinct vertices $x$ and $y$ are adjacent if and only if $A_{I}(xy)\neq A_{I}(x) \cup A_{I}(y)$, where $A_{I}(x) = \lbrace r\in R : rx\in I\rbrace$. In this paper, we study some basic properties of $AG_I(R)$, and we characterise when $AG_{I}(R)$ is planar, outerplanar or a ring graph. Also, we study the graph $AG_{I}(\mathbb{Z}_{n})$, where $Z_n$ is the ring of integers modulo $n$

Some results on the annihilator graph of a commutative ring

summary:Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace$. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \geq 1$

The Planar Index and Outerplanar Index of Some Graphs Associated to Commutative Rings

In this paper, we study the planar and outerplanar indices of some graphs associated to a commutative ring. We give a full characterization of these graphs with respect to their planar and outerplanar indices when R is a finite ring

Annihilator graphs of a commutative semigroup whose Zero-divisor graphs are a complete graph with an end vertex

Suppose that the zero-divisor graph of a commutative semi-group S, be a complete graph with an end vertex. In this paper, we determine the structure of the annihilator graph S and we show that if Z(S)= S, then the annihilator graph S is a disconnected graph

On the distinguishing chromatic number of the Kronecker products of graphs

AbstractIn this paper, we investigate the distinguishing chromatic number of Kronecker product of paths, cycles, star graphs, symmetric trees, almost symmetric trees, and bisymmetric trees