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- 저자 : Shariefuddin ) (검색결과 4 건)
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On graphs associated with modules over commutative rings
SHARIEFUDDIN PIRZADA,Rameez Raja 대한수학회 2016 대한수학회지 Vol.53 No.5
Let $M$ be an $R$-module, where $R$ is a commutative ring with identity $1$ and let $G(V,E)$ be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ to $M$ called full annihilating, semi-annihilating and star-annihilating graph. When $M$ is finite over $R$, we investigate metric dimensions in $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$. We show that $M$ over $R$ is finite if and only if the metric dimension of the graph $ann_f(\Gamma(M_R))$ is finite. We further show that the graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ are empty if and only if $M$ is a prime-multiplication-like $R$-module. We investigate the case when $M$ is a free $R$-module, where $R$ is an integral domain and show that the graphs $ann_f(\Gamma(M_R))$, $ann_s(\Gamma(M_R))$ and $ann_t(\Gamma(M_R))$ are empty if and only if $M \cong R$. Finally, we characterize all the non-simple weakly virtually divisible modules $M$ for which $Ann(M)$ is a prime ideal and $Soc(M) = 0$.
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ON GRAPHS ASSOCIATED WITH MODULES OVER COMMUTATIVE RINGS
Pirzada, Shariefuddin,Raja, Rameez Korean Mathematical Society 2016 대한수학회지 Vol.53 No.5
Let M be an R-module, where R is a commutative ring with identity 1 and let G(V,E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$. We show that M over R is finite if and only if the metric dimension of the graph $ann_f({\Gamma}(M_R))$ is finite. We further show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if M is a prime-multiplication-like R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs $ann_f({\Gamma}(M_R))$, $ann_s({\Gamma}(M_R))$ and $ann_t({\Gamma}(M_R))$ are empty if and only if $$M{\sim_=}R$$. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M) is a prime ideal and Soc(M) = 0.
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Mark Sequences and Sets in Directed Graphs
( Pirzada,Shariefuddin ) 한국수학교육학회 2011 수학교육 학술지 Vol.2011 No.-
Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science and technology. It is being actively used in fields as varied as biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). The powerful combinatorial methods found in graph theory have also been used to prove fundamental results in other areas of pure mathematics. One of the important theories is directed graphs, which represents the binary relations, and finds applications in many branches of sciences including computer sciences and networking. In directed graphs the concept of mark (score) attached to the vertex has been significantly investigated and used to study the structural properties. In this talk, we discuss various existence and constructive criterions for sequences of non-negative integers to be mark sequences and mark sets of various types of directed graphs. We also obtain algorithms for constructions of certain types of directed graphs.
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On strong metric dimension of zero-divisor graphs of rings
M. Imran Bhat,SHARIEFUDDIN PIRZADA 강원경기수학회 2019 한국수학논문집 Vol.27 No.3
In this paper, we study the strong metric dimension of zero-divisor graph $\Gamma(R)$ associated to a ring $R$. This is done by transforming the problem into a more well-known problem of finding the vertex cover number $\alpha(G)$ of a strong resolving graph $G_{sr}$. We find the strong metric dimension of zero-divisor graphs of the ring $\mathbb{Z}_n$ of integers modulo $n$ and the ring of Gaussian integers $\mathbb{Z}_n[i]$ modulo $n$. We obtain the bounds for strong metric dimension of zero-divisor graphs and we also discuss the strong metric dimension of the Cartesian product of graphs.
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