scholarly journals On the metric dimension and fractional metric dimension for hierarchical product of graphs

2013 ◽  
Vol 7 (2) ◽  
pp. 302-313 ◽  
Author(s):  
Min Feng ◽  
Kaishun Wang
Keyword(s):  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Product Of Graphs

A set of vertices W resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension for G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In order to study the metric dimension for the hierarchical product Gu22 ? Gu11 of two rooted graphs Gu22 and Gu11, we first introduce a new parameter, the rooted metric dimension rdim(Gu11) for a rooted graph Gu11. If G1 is not a path with an end-vertex u1, we show that dim(Gu22 ? Gu11) = |V(G2)|? rdim(Gu11), where |V(G2)| is the order of G2. If G1 is a path with an end-vertex u1, we obtain some tight inequalities for dim(Gu22 ? Gu11). Finally, we show that similar results hold for the fractional metric dimension.

scholarly journals On Metric Dimensions of Symmetric Graphs Obtained by Rooted Product

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 191 ◽  
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.

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 Resolvability in Subdivision of Circulant Networks Cn1,k

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.

scholarly journals 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 ◽  
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.

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 BI-DIMENSI METRIK DARI GRAF ANTIPRISMA

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. .

scholarly journals Dimensi Metrik Graf Kr+mKsr, m, r, s, En

CAUCHY ◽  
2011 ◽  
Vol 1 (4) ◽  
pp. 165
Author(s):  
Hindayani Hindayani
Keyword(s):  
Metric Dimension ◽  
Combinatorial Search ◽  
Robotic Navigation ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Metric Dimensions ◽  
Partition Graph

<div class="standard"><a id="magicparlabel-29">The concept of minimum resolving set has proved to be useful and or related to a variety of fields such as Chemistry, Robotic Navigation, and Combinatorial Search and Optimization. So that, this thesis explains the metric dimension of graph Kr + mKsr, m, r, s E N. Resolving set of a graph G is a subset of F (G) that its distance representation is distinct to all vertices of graph G. Resolving set with minimum cardinality is called minimum resolving set, and cardinal states metric dimension of G and noted with dim (G). By drawing the graph, it will be found the resolving set, minimum resolving set and the metric dimension easily. After that, formulate those metric dimensions into a theorem. This research search for the metric dimension of Kr + mKs, m &gt; 2, m,r,s E N and its outcome are dim (Kr + mK1)= m+ (r-2) and dim(Kr + mKs)= m(s-1)+(r-2). This research can be continued for determining the metric dimension of another graph, by changing the operation of its graph or partition graph.</a></div>

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 On Resolvability Parameters of Some Wheel-Related Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Bin Yang ◽  
Muhammad Rafiullah ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Sarfraz Ahmad
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Hard Problem ◽  
Np Hard ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Np Hard Problem ◽  
Simple Connected Graph

Let G=V,E be a simple connected graph, w∈V be a vertex, and e=uv∈E be an edge. The distance between the vertex w and edge e is given by de,w=mindw,u,dw,v, A vertex w distinguishes two edges e1, e2∈E if dw,e1≠dw,e2. A set S is said to be resolving if every pair of edges of G is distinguished by some vertices of S. A resolving set with minimum cardinality is the basis for G, and this cardinality is the edge metric dimension of G, denoted by edimG. It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

scholarly journals The Vertex-Edge Resolvability of Some Wheel-Related Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Bao-Hua Xing ◽  
Sunny Kumar Sharma ◽  
Vijay Kumar Bhat ◽  
Hassan Raza ◽  
Jia-Bao Liu
Keyword(s):  
Connected Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Wheel Graph ◽  
Wheel Graphs ◽  
Metric Dimensions ◽  
Mixed Metric

A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m  dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

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