Edge metric dimension of some classes of circulant graphs

Muhammad Ahsan; Zohaib Zahid; Sohail Zafar

doi:10.2478/auom-2020-0032

Edge metric dimension of some classes of circulant graphs

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2020-0032 ◽

2020 ◽

Vol 28 (3) ◽

pp. 15-37

Author(s):

Muhammad Ahsan ◽

Zohaib Zahid ◽

Sohail Zafar

Keyword(s):

Connected Graph ◽

Ordered Set ◽

Metric Dimension ◽

Circulant Graph ◽

Circulant Graphs ◽

Minimum Cardinality ◽

Vertex Set ◽

Metric Basis

AbstractLet G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

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.

Fault-Tolerant Metric Dimension of Circulant Graphs

Mathematics ◽

10.3390/math10010124 ◽

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

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1904781s ◽

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 $C=\lbrace \pm 1,\pm 2\rbrace$.

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.

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.

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.

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.

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.

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.

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.