scholarly journals Computation of Edge Resolvability of Benzenoid Tripod Structure

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ali Ahmad ◽  
Sadia Husain ◽  
Muhammad Azeem ◽  
Kashif Elahi ◽  
M. K. Siddiqui
Keyword(s):  
Fault Tolerant ◽  
Pharmaceutical Research ◽  
Chemical Compounds ◽  
Metric Dimension ◽  
Resolving Set ◽  
Unique Representation ◽  
Chemical Structures ◽  
Distance Vector

In chemistry, graphs are commonly used to show the structure of chemical compounds, with nodes and edges representing the atom and bond types, respectively. Edge resolving set λ e is an ordered subset of nodes of a graph C , in which each edge of C is distinctively determined by its distance vector to the nodes in λ . The cardinality of a minimum edge resolving set is called the edge metric dimension of C . An edge resolving set L e , f of C is fault-tolerant if λ e , f ∖ b is also an edge resolving set, for every b in λ e , f . Resolving set allows obtaining a unique representation for chemical structures. In particular, they were used in pharmaceutical research for discovering patterns common to a variety of drugs. In this paper, we determine the exact edge metric and fault-tolerant edge metric dimension of benzenoid tripod structure and proved that both parameters are constant.

scholarly journals Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 124
Author(s):  
Laxman Saha ◽  
Rupen Lama ◽  
Kalishankar Tiwary ◽  
Kinkar Chandra Das ◽  
Yilun Shang
Keyword(s):  
Connected Graph ◽  
Fault Tolerant ◽  
Metric Dimension ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set

Let G be a connected graph with vertex set V(G) and d(u,v) be the distance between the vertices u and v. A set of vertices S={s1,s2,…,sk}⊂V(G) is called a resolving set for G if, for any two distinct vertices u,v∈V(G), there is a vertex si∈S such that d(u,si)≠d(v,si). A resolving set S for G is fault-tolerant if S\{x} is also a resolving set, for each x in S, and the fault-tolerant metric dimension of G, denoted by β′(G), is the minimum cardinality of such a set. The paper of Basak et al. on fault-tolerant metric dimension of circulant graphs Cn(1,2,3) has determined the exact value of β′(Cn(1,2,3)). In this article, we extend the results of Basak et al. to the graph Cn(1,2,3,4) and obtain the exact value of β′(Cn(1,2,3,4)) for all n≥22.

scholarly journals Edge Metric and Fault-Tolerant Edge Metric Dimension of Hollow Coronoid

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1405
Author(s):  
Ali N. A. Koam ◽  
Ali Ahmad ◽  
Muhammad Ibrahim ◽  
Muhammad Azeem
Keyword(s):  
Graph Theory ◽  
Chemical Structure ◽  
Fault Tolerant ◽  
Organic Chemical ◽  
Metric Dimension ◽  
Chemical Structures ◽  
Geometric Arrangements

Geometric arrangements of hexagons into six sides of benzenoids are known as coronoid systems. They are organic chemical structures by definition. Hollow coronoids are divided into two types: primitive and catacondensed coronoids. Polycyclic conjugated hydrocarbon is another name for them. Chemical mathematics piques the curiosity of scientists from a variety of disciplines. Graph theory has always played an important role in making chemical structures intelligible and useful. After converting a chemical structure into a graph, many theoretical and investigative studies on structures can be carried out. Among the different parameters of graph theory, the dimension of edge metric is the most recent, unique, and important parameter. Few proposed vertices are picked in this notion, such as all graph edges have unique locations or identifications. Different (edge) metric-based concept for the structure of hollow coronoid were discussed in this study.

scholarly journals Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Zhi-Bo Zheng ◽  
Ashfaq Ahmad ◽  
Zaffar Hussain ◽  
Mobeen Munir ◽  
Muhammad Imran Qureshi ◽  
...  
Keyword(s):  
Present Article ◽  
Open Problem ◽  
Ordered Set ◽  
Fault Tolerant ◽  
Convex Polytopes ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Wheel Graph

For a graph G , an ordered set S ⊆ V G is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V G . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v is still the resolving set ∀ v ∈ V G . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6 . We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.

scholarly journals FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS

2019 ◽  
pp. 781
Author(s):  
Narjes Seyedi ◽  
Hamid Reza Maimani
Keyword(s):  
Fault Tolerant ◽  
Additive Group ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set

A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(G)$. A resolving set $W'$ for $G$ is fault-tolerant if $W'\setminus \left\lbrace w\right\rbrace $ for each $w$ in $W'$, is also a resolving set and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set, denoted by $\beta'(G)$. The circulant graph is a graph with vertex set $\mathbb{Z}_{n}$, an additive group of integers modulo $n$, and two vertices labeled $i$ and $j$ adjacent if and only if $i -j \left( mod \ n \right)  \in C$, where $C \in \mathbb{Z}_{n}$ has the property that $C=-C$ and $0 \notin C$. The circulant graph is denoted by $X_{n,\bigtriangleup}$ where $\bigtriangleup = \vert C\vert$. In this paper, we study the fault-tolerant metric dimension of a family of circulant graphs $X_{n,3}$ with connection set $C=\lbrace 1,\dfrac{n}{2},n-1\rbrace$ and circulant graphs $X_{n,4}$ with connection set $C=\lbrace \pm 1,\pm 2\rbrace$.

scholarly journals On Fault-Tolerant Resolving Sets of Some Families of Ladder Networks

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hua Wang ◽  
Muhammad Azeem ◽  
Muhammad Faisal Nadeem ◽  
Ata Ur-Rehman ◽  
Adnan Aslam
Keyword(s):  
Computer Networks ◽  
Computer Network ◽  
Fault Tolerant ◽  
Metric Dimension ◽  
Resolving Set ◽  
Arbitrary Vertex

In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in different areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension differ by 1.

scholarly journals Fault-Tolerant Resolvability in Some Classes of Line Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Xuan Guo ◽  
Muhammad Faheem ◽  
Zohaib Zahid ◽  
Waqas Nazeer ◽  
Jingjng Li
Keyword(s):  
Graph Theory ◽  
Fault Tolerance ◽  
Fault Tolerant ◽  
Stable System ◽  
Subject Classification ◽  
Metric Dimension ◽  
Line Graphs ◽  
Resolving Set ◽  
Minimum Cardinality

Fault tolerance is the characteristic of a system that permits it to carry on its intended operations in case of the failure of one of its units. Such a system is known as the fault-tolerant self-stable system. In graph theory, if we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this paper, we determine the fault-tolerant resolvability in line graphs. As a main result, we computed the fault-tolerant metric dimension of line graphs of necklace and prism graphs (2010 Mathematics Subject Classification: 05C78).

Therapeutic Properties of Several Chemical Compounds from Salvia officinalis L. in Alzheimer’s Disease

2020 ◽  
Vol 21 ◽  
Author(s):  
Georgiana Uță ◽  
Denisa Ștefania Manolescu ◽  
Speranța Avram
Keyword(s):  
Alzheimer’S Disease ◽  
Alzheimer's Disease ◽  
Side Effects ◽  
Chemical Compounds ◽  
Natural Compounds ◽  
Salvia Officinalis ◽  
Chemical Structures ◽  
In Silico Studies ◽  
Wide Range ◽  
Salvia Officinalis L

Background.: Currently, the pharmacological management in Alzheimer's disease is based on several chemical structures, represented by acetylcholinesterase and N-methyl-D-aspartate (NMDA) receptor ligands, with still unclear molecular mechanisms, but severe side effects. For this reason, a challenge for Alzheimer's disease treatment remains to identify new drugs with reduced side effects. Recently, the natural compounds, in particular certain chemical compounds identified in the essential oil of peppermint, sage, grapes, sea buckthorn, have increased interest as possible therapeutics. Objectives.: In this paper, we have summarized data from the recent literature, on several chemical compounds extracted from Salvia officinalis L., with therapeutic potential in Alzheimer's disease. Methods.: In addition to the wide range of experimental methods performed in vivo and in vitro, also we presented some in silico studies of medicinal compounds. Results. Through this mini-review, we present the latest information regarding the therapeutic characteristics of natural compounds isolated from Salvia officinalis L. in Alzheimer's disease. Conclusion.: Thus, based on the information presented, we can say that phytotherapy is a reliable therapeutic method in a neurodegenerative disease.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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