On Metric Dimension and Fault Tolerant Metric Dimension of Some Chemical Structures

2021 ◽  
pp. 1-13
Author(s):  
Muhammad Faisal Nadeem ◽  
Ayesha Shabbir ◽  
Muhammad Azeem
Keyword(s):  
Fault Tolerant ◽  
Metric Dimension ◽  
Chemical Structures

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 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.

On the fault-tolerant metric dimension of certain interconnection networks

2018 ◽  
Vol 60 (1-2) ◽  
pp. 517-535 ◽  
Author(s):  
Hassan Raza ◽  
Sakander Hayat ◽  
Xiang-Feng Pan
Keyword(s):  
Interconnection Networks ◽  
Fault Tolerant ◽  
Metric Dimension

scholarly journals Fault-Tolerant Resolvability and Extremal Structures of Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 78 ◽  
Author(s):  
Hassan Raza ◽  
Sakander Hayat ◽  
Muhammad Imran ◽  
Xiang-Feng Pan
Keyword(s):  
Fault Tolerant ◽  
Upper Bounds ◽  
Metric Dimension ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Constant Fault

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n, n − 1 , and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure.

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.

Fault-tolerant metric dimension of circulant graphs C(1,2,3)

2020 ◽  
Vol 817 ◽  
pp. 66-79 ◽  
Author(s):  
Mithun Basak ◽  
Laxman Saha ◽  
Gour Kanta Das ◽  
Kalishankar Tiwary
Keyword(s):  
Fault Tolerant ◽  
Metric Dimension ◽  
Circulant Graphs

scholarly journals Fault-Tolerant Metric Dimension of Interconnection Networks

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 145435-145445
Author(s):  
Sakander Hayat ◽  
Asad Khan ◽  
Muhammad Yasir Hayat Malik ◽  
Muhammad Imran ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Interconnection Networks ◽  
Fault Tolerant ◽  
Metric Dimension
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