scholarly journals The k-Metric Dimension of a Unicyclic Graph

Mathematics ◽  
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.

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

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 300 ◽  
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.

scholarly journals Computing Edge Metric Dimension of One-Pentagonal Carbon Nanocone

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.

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

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

<p>Let <span class="math"><em>G</em></span> be a connected graph and let <span class="math"><em>u</em>, <em>v</em></span> <span class="math"> ∈ </span> <span class="math"><em>V</em>(<em>G</em>)</span>. For an ordered set <span class="math"><em>W</em> = {<em>w</em><sub>1</sub>, <em>w</em><sub>2</sub>, ..., <em>w</em><sub><em>n</em></sub>}</span> of <span class="math"><em>n</em></span> distinct vertices in <span class="math"><em>G</em></span>, the representation of a vertex <span class="math"><em>v</em></span> of <span class="math"><em>G</em></span> with respect to <span class="math"><em>W</em></span> is the <span class="math"><em>n</em></span>-vector <span class="math"><em>r</em>(<em>v</em>∣<em>W</em>) = (<em>d</em>(<em>v</em>, <em>w</em><sub>1</sub>), <em>d</em>(<em>v</em>, <em>w</em><sub>2</sub>), ..., </span> <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>n</em></sub>))</span>, where <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>i</em></sub>)</span> is the distance between <span class="math"><em>v</em></span> and <span class="math"><em>w</em><sub><em>i</em></sub></span> for <span class="math">1 ≤ <em>i</em> ≤ <em>n</em></span>. The set <span class="math"><em>W</em></span> is a local metric set of <span class="math"><em>G</em></span> if <span class="math"><em>r</em>(<em>u</em> ∣ <em>W</em>) ≠ <em>r</em>(<em>v</em> ∣ <em>W</em>)</span> for every pair <span class="math"><em>u</em>, <em>v</em></span> of adjacent vertices of <span class="math"><em>G</em></span>. The local metric set of <span class="math"><em>G</em></span> with minimum cardinality is called a local metric basis for <span class="math"><em>G</em></span> and its cardinality is called a local metric dimension, denoted by <span class="math"><em>l</em><em>m</em><em>d</em>(<em>G</em>)</span>. In this paper we determine the local metric dimension of a <span class="math"><em>t</em></span>-fold wheel graph, <span class="math"><em>P</em><sub><em>n</em></sub></span> <span class="math"> ⊙ </span> <span class="math"><em>K</em><sub><em>m</em></sub></span> graph, and generalized fan graph.</p>

scholarly journals Edge metric dimension of some classes of circulant graphs

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.

Bounds on the sum of domination number and metric dimension of graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Cong X. Kang ◽  
Eunjeong Yi
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Domination Number ◽  
Unicyclic Graph ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
Dimension Formula ◽  
Vertex Set ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text]. The domination number, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set [Formula: see text] such that every vertex not in [Formula: see text] is adjacent to a vertex in [Formula: see text]. The metric dimension, [Formula: see text], of [Formula: see text] is the minimum cardinality of a set of vertices such that every vertex of [Formula: see text] is uniquely determined by its vector of distances to the chosen vertices. For a tree [Formula: see text] of order at least two, we show that [Formula: see text], where [Formula: see text] denotes the number of exterior major vertices of [Formula: see text]; further, we characterize trees [Formula: see text] achieving equality. For a connected graph [Formula: see text] of order [Formula: see text], Bagheri Gh. et al. proved that [Formula: see text] and equality holds if and only if [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the complete graph and [Formula: see text] denotes a complete bi-partite graph of order [Formula: see text]. We characterize graphs [Formula: see text] for which [Formula: see text] equals two and three, respectively. We also characterize graphs [Formula: see text] satisfying [Formula: see text] when [Formula: see text] is a tree, a unicyclic graph, or a complete multi-partite graph.

On the Strong Metric Dimension of Certain Nanostructures

2017 ◽  
Vol 14 (1) ◽  
pp. 354-358 ◽  
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.

scholarly journals A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi–objective linear programming problem

2017 ◽  
Vol 27 (3) ◽  
pp. 563-573 ◽  
Author(s):  
Rajendran Vidhya ◽  
Rajkumar Irene Hepzibah
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Method ◽  
Arithmetic Operations ◽  
Intuitionistic Fuzzy Numbers ◽  
Multi Objective ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

AbstractIn a real world situation, whenever ambiguity exists in the modeling of intuitionistic fuzzy numbers (IFNs), interval valued intuitionistic fuzzy numbers (IVIFNs) are often used in order to represent a range of IFNs unstable from the most pessimistic evaluation to the most optimistic one. IVIFNs are a construction which helps us to avoid such a prohibitive complexity. This paper is focused on two types of arithmetic operations on interval valued intuitionistic fuzzy numbers (IVIFNs) to solve the interval valued intuitionistic fuzzy multi-objective linear programming problem with pentagonal intuitionistic fuzzy numbers (PIFNs) by assuming differentαandβcut values in a comparative manner. The objective functions involved in the problem are ranked by the ratio ranking method and the problem is solved by the preemptive optimization method. An illustrative example with MATLAB outputs is presented in order to clarify the potential approach.

Sign in / Sign up

Export Citation Format

Share Document