Browsing by Author "Pejman, S. Batool"

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Access status: Open Access ,
Metric dimension of Andrásfai graphs
(Wydawnictwa AGH, 2019) Pejman, S. Batool; Payrovi, Shiroyeh; Behtoei, Ali
A set $W\subseteq V(G)$ is called a resolving set, if for each pair of distinct vertices $u,v \in V(G)$ there exists $t \in W$ such that $d(u,t)\neq d(v,t)$, where $d(x,y)$ is the distance between vertices $x$ and $y$. The cardinality of a minimum resolving set for $G$ is called the metric dimension of $G$ and is denoted by $dim_{M}(G)$. This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and $And(k)\square P_n$. Also, we provide upper and lower bounds for $dim_M(And(k)\square C_n)$.
Access status: Open Access ,
The intersection graph of annihilator submodules of a module
(Wydawnictwa AGH, 2019) Pejman, S. Batool; Payrovi, Shiroyeh; Babaei, Sakineh
Let $R$ be a commutative ring and $M$ be a Noetherian $R$-module. The intersection graph of annihilator submodules of $M$, denoted by $GA(M)$ is an undirected simple graph whose vertices are the classes of elements of $Z_R(M)\setminus \text{Ann}_R(M)$, for $a,b \in R$ two distinct classes $[a]$ and $[b]$ are adjacent if and only if $\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0$. In this paper, we study diameter and girth of $GA(M)$ and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that $GA(M)$ is complete if and only if $Z_{R}(M)$ is an ideal of $R$. Also, we show that if $M$ is a finitely generated $R$-module with $r(\text{Ann}_R(M))\neq \text{Ann}_R(M)$ and $|m-\text{Ass}_R(M)|=1$ and $GA(M)$ is a star graph, then $r(\text{Ann}_{R}(M))$ is not a prime ideal of $R$ and $|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1$.