Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

Frontiers in Physics ◽

10.3389/fphy.2021.749166 ◽

2021 ◽

Vol 9 ◽

Author(s):

Sunny Kumar Sharma ◽

Hassan Raza ◽

Vijay Kumar Bhat

Keyword(s):

Combinatorial Chemistry ◽

Connected Graph ◽

Metric Dimension ◽

Molecular Topology ◽

Resolving Set ◽

Minimum Cardinality ◽

Crucial Information ◽

Metric Basis

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.

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Metric basis and metric dimension of 1-pentagonal carbon nanocone networks

Scientific Reports ◽

10.1038/s41598-020-76516-1 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Ashfaq Ahmad ◽

Maqbool Chaudhary ◽

Junaid Alam Khan ◽

...

Keyword(s):

Combinatorial Chemistry ◽

Connected Graph ◽

Cardinal Number ◽

Metric Dimension ◽

Molecular Topology ◽

Resolving Set ◽

Computer Chemistry ◽

Carbon Nanocones ◽

Vertex Set ◽

Metric Basis

AbstractResolving set and metric basis has become an integral part in combinatorial chemistry and molecular topology. It has a lot of applications in computer, chemistry, pharmacy and mathematical disciplines. A subset S of the vertex set V of a connected graph G resolves G if all vertices of G have different representations with respect to S. A metric basis for G is a resolving set having minimum cardinal number and this cardinal number is called the metric dimension of G. In present work, we find a metric basis and also metric dimension of 1-pentagonal carbon nanocones. We conclude that only three vertices are minimal requirement for the unique identification of all vertices in this network.

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Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes

Symmetry ◽

10.3390/sym10080300 ◽

2018 ◽

Vol 10 (8) ◽

pp. 300 ◽

Cited By ~ 5

Author(s):

Zafar Hussain ◽

Mobeen Munir ◽

Maqbool Chaudhary ◽

Shin Kang

Keyword(s):

Electronic Structure ◽

Transport Properties ◽

Present Article ◽

Connected Graph ◽

Metric Dimension ◽

Lattice Structures ◽

Resolving Set ◽

Minimum Cardinality ◽

Mathematical Sciences ◽

Metric Basis

Concepts of resolving set and metric basis has enjoyed a lot of success because of multi-purpose applications both in computer and mathematical sciences. For a connected graph G(V,E) a subset W of V(G) is a resolving set for G if every two vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a metric basis for graph G and this minimum cardinality is known as metric dimension of G. Boron nanotubes with different lattice structures, radii and chirality’s have attracted attention due to their transport properties, electronic structure and structural stability. In the present article, we compute the metric dimension and metric basis of 2D lattices of alpha-boron nanotubes.

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On the Strong Metric Dimension of Certain Nanostructures

Journal of Computational and Theoretical Nanoscience ◽

10.1166/jctn.2017.6328 ◽

2017 ◽

Vol 14 (1) ◽

pp. 354-358 ◽

Cited By ~ 1

Author(s):

Sathish Krishnan ◽

Bharati Rajan ◽

Muhammad Imran

Keyword(s):

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Strong Metric Dimension ◽

Metric Basis

Let G(V, E) be a connected graph. A vertex w strongly resolves a pair of vertices u,v in V if there exists some shortestu–w path containing V or some shortest v–w path containing u. A set w ⊂ V of vertices is called a strong resolving set for G if every pair of vertices of V\W is strongly resolved by some vertex of w . A strong resolving set of minimum cardinality is called a strong metric basis and this cardinality is called the strong metric dimension of G. The strong metric dimension problem is to find a strong metric basis in a graph. In this paper we investigate the strong metric dimension problem for certain nanostructures.

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On the metric dimension and diameter of circulant graphs with three jumps

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500088 ◽

2018 ◽

Vol 10 (01) ◽

pp. 1850008

Author(s):

Muhammad Imran ◽

A. Q. Baig ◽

Saima Rashid ◽

Andrea Semaničová-Feňovčíková

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Metric Spaces ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Infinite Class ◽

Metric Properties ◽

Minimum Number ◽

Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

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On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽

10.3390/math6100191 ◽

2018 ◽

Vol 6 (10) ◽

pp. 191 ◽

Cited By ~ 4

Author(s):

Shahid Imran ◽

Muhammad Siddiqui ◽

Muhammad Imran ◽

Muhammad Hussain

Keyword(s):

Connected Graph ◽

The Other ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Four Dimensions ◽

Symmetric Graphs ◽

Metric Dimensions ◽

Cycle Path

Let G = (V, E) be a connected graph and d(x, y) be the distance between the vertices x and y in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). In this paper, Cycle, Path, Harary graphs and their rooted product as well as their connectivity are studied and their metric dimension is calculated. It is proven that metric dimension of some graphs is unbounded while the other graphs are constant, having three or four dimensions in certain cases.

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Resolvability in Subdivision of Circulant Networks Cn1,k

Discrete Dynamics in Nature and Society ◽

10.1155/2020/4197678 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Jianxin Wei ◽

Syed Ahtsham Ul Haq Bokhary ◽

Ghulam Abbas ◽

Muhammad Imran

Keyword(s):

Facility Location ◽

Connected Graph ◽

Location Problems ◽

Metric Dimension ◽

Circulant Graph ◽

Resolving Set ◽

Minimum Cardinality ◽

Facility Location Problems ◽

Wide Range ◽

Circulant Networks

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

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Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽

10.1155/2020/6267072 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8

Author(s):

Jia-Bao Liu ◽

Ali Zafari

Keyword(s):

Connected Graph ◽

Line Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality ◽

Strong Metric Dimension ◽

Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

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The k-Metric Dimension of a Unicyclic Graph

Mathematics ◽

10.3390/math9212789 ◽

2021 ◽

Vol 9 (21) ◽

pp. 2789

Author(s):

Alejandro Estrada-Moreno

Keyword(s):

Linear Programming ◽

Programming Problem ◽

Connected Graph ◽

Linear Programming Problem ◽

Unicyclic Graph ◽

Metric Dimension ◽

Minimum Cardinality ◽

Metric Basis

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

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On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2018.2.2.4 ◽

2018 ◽

Vol 2 (2) ◽

pp. 88

Author(s):

Rokhana Ayu Solekhah ◽

Tri Atmojo Kusmayadi

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Minimum Cardinality ◽

Wheel Graph ◽

Metric Set ◽

Metric Basis ◽

Fan Graph

Let G be a connected graph and let u, v ∈ V(G). For an ordered set W = {w1, w2, ..., wn} of n distinct vertices in G, the representation of a vertex v of G with respect to W is the n-vector r(v∣W) = (d(v, w1), d(v, w2), ..., d(v, wn)), where d(v, wi) is the distance between v and wi for 1 ≤ i ≤ n. The set W is a local metric set of G if r(u ∣ W) ≠ r(v ∣ W) for every pair u, v of adjacent vertices of G. The local metric set of G with minimum cardinality is called a local metric basis for G and its cardinality is called a local metric dimension, denoted by lmd(G). In this paper we determine the local metric dimension of a t-fold wheel graph, Pn ⊙ Km graph, and generalized fan graph.

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BI-DIMENSI METRIK DARI GRAF ANTIPRISMA

Majalah Ilmiah Matematika dan Statistika ◽

10.19184/mims.v20i2.19639 ◽

2020 ◽

Vol 20 (2) ◽

pp. 53

Author(s):

Hendy Hendy ◽

M. Ismail Marzuki

Keyword(s):

Shortest Path ◽

Connected Graph ◽

Metric Dimension ◽

Resolving Set ◽

Minimum Cardinality

Let G = (V, E) be a simple and connected graph. For each x ∈ V(G), it is associated with a vector pair (a, b), denoted by S x , corresponding to subset S = {s1 , s2 , ... , s k } ⊆ V(G), with a = (d(x, s1 ), d(x, s2 ), ... , d(x, s k )) and b = (δ(x, s1 ), δ(x, s2 ), ... , δ(x, s k )). d(v, s) is the length of shortest path from vertex v to s, and δ(v, s) is the length of the furthest path from vertex v to s. The set S is called the bi-resolving set in G if S x ≠ S y for any two distinct vertices x, y ∈ V(G). The bi- metric dimension of graph G, denoted by β b (G), is the minimum cardinality of the bi-resolving set in graph G. In this study we analyze bi-metric dimension in the antiprism graph (A n ). From the analysis that has been done, it is obtained the result that bi-metric dimension of graph A n , β b (A n ) is 3. Keywords: Antiprism graph, bi-metric dimension, bi-resolving set. .

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